#72233397

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	C₂×#2021
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁴
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

4213 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:

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

Polycyclic presentation:

2^10|a^2=ij,c^2=j,e^2=i,c^a=ci,c^b=cg,d^a=di,d^b=df,d^c=de,e^b=eh,e^c=ei,e^d=ei,f^c=fhij,f^e=fj,g^d=gi,h^d=hj

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233401, #72234085, #72234092
maximal normal subgroup of	#72233401, #72234085, #72234092

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