#72287629

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	#33758	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×#322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2264 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)(4,17)(5,10)(6,11)(7,12)(9,20)(13,26)(14,23)(15,24)(16,25)(18,22)(19,28)(21,27)(29,31)(30,32)(33,34)(35,43)(36,41)(37,42)(38,39)(40,47)(44,52)(45,50)(46,57)(48,56)(49,51)(53,62)(54,59)(55,61)(58,63)(60,64);(1,3)(2,8)(4,18)(5,15)(6,16)(7,14)(9,26)(10,24)(11,25)(12,23)(13,20)(17,22)(19,29)(21,30)(27,32)(28,31)(33,34)(35,43)(36,41)(37,42)(38,39)(40,47)(44,52)(45,50)(46,57)(48,56)(49,51)(53,62)(54,59)(55,61)(58,63)(60,64);(1,4,6,19,7,20,21,5)(2,9,11,10,12,17,27,28)(3,13,16,15,14,18,30,29)(8,22,25,31,23,26,32,24)(34,39)(35,44,47,59)(36,48)(37,49)(40,52,43,54)(41,55)(45,61)(46,58,62,60)(50,56)(53,63,57,64);(3,14)(8,23)(13,18)(15,29)(16,30)(22,26)(24,31)(25,32)(33,35,45,59,37,46,61,58,38,47,36,44,49,62,48,60)(34,40,50,52,42,53,55,64,39,43,41,54,51,57,56,63);(1,5,21,20,7,19,6,4)(2,10,27,9,12,28,11,17)(3,15,30,13,14,29,16,18)(8,24,32,26,23,31,25,22)(35,46,47,62)(36,45)(40,53,43,57)(41,50)(44,58,59,60)(48,61)(52,63,54,64)(55,56)

Polycyclic presentation:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287659, #72287829, #72343446
maximal normal subgroup of	#72287659, #72287829, #72343446

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