#72233495

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23900
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁴
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#121 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

4370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(3,13,18,26)(4,19)(9,25,23,27)(10,29)(14,28,32,31)(15,16,21,17)(20,22)(24,30);(1,2,5,12)(3,14,18,32)(4,10,19,29)(6,11,7,8)(9,26,23,13)(15,28,21,31)(16,25,17,27)(20,30,22,24);(2,8)(3,15,18,21)(4,20)(9,27,23,25)(10,29)(11,12)(13,16,26,17)(14,31,32,28)(19,22)(24,30);(1,3)(2,9)(4,21)(5,17)(6,16)(7,18)(8,23)(10,14)(11,31)(12,28)(13,20)(15,22)(19,26)(24,32)(25,30)(27,29)

Polycyclic presentation:

2^10|a^2=hij,b^2=ghj,c^2=hij,e^2=ij,b^a=bg,c^b=cj,d^a=de,d^b=deij,d^c=def,e^a=ei,e^c=eh,e^d=eij,f^a=fhij,f^b=fhj,f^c=fhij,f^d=fj,f^e=fj,g^d=gi,h^d=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233500
maximal normal subgroup of	#72233500

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