#72287547

order	1024
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	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623514	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×#322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|c^2=g,d^2=f,e^2=fh,f^2=h,g^2=i,i^2=j,c^a=ch,d^a=dhj,d^b=dhj,d^c=dghi,e^c=eij,f^c=fj,g^d=gi,g^e=gj,i^d=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287605, #72287829, #72287851, …
maximal normal subgroup of	#72287605, #72287829, #72287851, …

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.6*F.8);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);