#72287795

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂³×#97
prime radical	C₂²×#41	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2525 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)(3,7)(4,8)(5,10)(6,9)(11,17)(12,18)(13,19)(14,21)(15,20)(16,22)(23,27)(24,28)(25,29)(26,30)(31,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,17)(9,18)(10,19)(14,24)(15,23)(16,25)(20,27)(21,28)(22,29)(26,31)(30,32)(33,34)(35,43)(36,41)(37,42)(38,39)(40,45)(44,53)(46,57)(47,56)(48,51)(49,55)(50,52)(54,60)(58,62)(59,63)(61,64);(1,4,5,14,6,15,16,26)(2,8,10,21,9,20,22,30)(3,11,13,24,12,23,25,31)(7,17,19,28,18,27,29,32)(34,39)(35,44,45,47)(36,48)(40,53,43,56)(42,52)(46,61,60,59)(49,58)(54,64,57,63);(3,12)(7,18)(11,23)(13,25)(17,27)(19,29)(24,31)(28,32);(2,9)(3,12)(8,20)(10,22)(11,23)(13,25)(21,30)(24,31)(33,35,36,46,37,47,49,61,38,45,48,60,50,44,58,59)(34,40,41,54,42,53,55,63,39,43,51,57,52,56,62,64)

Polycyclic presentation:

2^10|c^2=f,e^2=g,f^2=h,g^2=i,i^2=j,c^b=cj,d^b=dh,e^a=eh,e^b=ehj,e^c=eij,f^e=fj,g^c=gj

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287811, #72318591, #72320746
maximal normal subgroup of	#72287811, #72318591, #72320746

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