#72318525

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	C₂×#466	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2198 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)(3,8)(4,15)(5,10)(6,11)(7,12)(9,17)(13,21)(14,22)(16,28)(18,24)(19,25)(20,26)(23,29)(27,31)(30,32);(1,3,6,4,7,14,18,17)(2,8,11,15,12,22,24,9)(5,13,20,16,19,27,30,29)(10,21,26,28,25,31,32,23)(33,34,37,43,39,45,40,56)(35,46,48,63,50,57,51,60)(36,42,53,61,54,44,38,55)(41,47,58,49,59,64,62,52);(1,4,6,14,7,17,18,3)(2,9,11,8,12,15,24,22)(5,16,20,27,19,29,30,13)(10,23,26,21,25,28,32,31)(33,35)(34,41,45,59)(36,52)(37,51)(38,47)(39,50)(40,48)(42,60,44,63)(43,62,56,58)(46,61,57,55)(49,54)(53,64);(1,5,18,30,7,19,6,20)(2,10,24,32,12,25,11,26)(3,13,17,29,14,27,4,16)(8,21,9,23,22,31,15,28)(33,36,40,38,39,54,37,53)(34,42,56,55,45,44,43,61)(35,47,51,52,50,64,48,49)(41,57,62,63,59,46,58,60);(5,19)(10,25)(13,27)(16,29)(20,30)(21,31)(23,28)(26,32)

Polycyclic presentation:

2^10|b^2=f,c^2=fi,d^2=fh,f^2=h,g^2=i,i^2=j,c^a=chj,c^b=cg,d^c=di,e^d=ehj,f^c=fj,g^b=gi,g^c=gi,g^d=gj,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318591, #72318672, #72319496
maximal normal subgroup of	#72318591, #72318672, #72319496

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