#72287776

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×#442
commutator	C₄×C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁵
prime radical	#72287577	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #184046 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2,9,12)(3,13,19,31)(4,20,24,6)(5,25,27,23)(7,11,8,22)(10,17,30,18)(14,28,32,16)(15,29,26,21)(33,34,40,44)(35,39,43,38)(36,41,52,60)(37,42,55,62)(45,56,47,50)(46,54,61,53)(48,57,49,58)(51,59,64,63);(1,3,8,18,9,19,7,17)(2,10,11,13,12,30,22,31)(4,21,23,32,24,29,25,14)(5,26,20,16,27,15,6,28)(33,35)(34,39)(36,47)(37,46)(38,44)(40,43)(41,50)(42,54)(45,52)(48,63)(49,59)(51,57)(53,62)(55,61)(56,60)(58,64);(1,4,7,23,9,24,8,25)(2,6,11,27,12,20,22,5)(3,14,17,21,19,32,18,29)(10,15,13,16,30,26,31,28)(33,36,38,50,40,52,39,56)(34,41,35,45,44,60,43,47)(37,49,53,64,55,48,54,51)(42,58,46,63,62,57,61,59);(1,5,7,6,9,27,8,20)(2,4,22,25,12,24,11,23)(3,15,17,16,19,26,18,28)(10,29,31,32,30,21,13,14)(35,43)(36,48)(37,53)(38,39)(41,57)(42,46)(45,63)(47,59)(49,52)(50,64)(51,56)(54,55)(58,60)(61,62);(1,6,7,27,9,20,8,5)(2,4,11,23,12,24,22,25)(3,16,17,26,19,28,18,15)(10,29,13,14,30,21,31,32)(33,37,38,53,40,55,39,54)(34,42,35,46,44,62,43,61)(36,49,50,64,52,48,56,51)(41,58,45,63,60,57,47,59)

Polycyclic presentation:

2^11|a^2=j,b^2=h,c^2=f,d^2=hjk,e^2=f,f^2=j,g^2=i,h^2=jk,i^2=k,b^a=bh,c^a=cjk,c^b=chk,d^a=dhjk,d^b=dj,d^c=dg,e^b=ejk,e^d=eik,f^d=fk,g^a=gjk,g^c=gi,g^d=gij,g^e=gk,h^a=hjk,h^c=hjk,i^c=ik,i^d=ik

Conjugacy classes & character table

131 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.8*F.10*F.11);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10*F.11);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.10*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.10*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 9, 4, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);