#72235585

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×#95
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂²	C₂³×#95
prime radical	C₂⁶	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

11505 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=g,e^2=i,f^2=i,h^2=ij,b^a=bh,c^a=ci,c^b=cj,d^a=dij,d^b=di,e^b=ej,f^b=fi,g^a=gij,h^a=hj

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235632, #72235672, #72235683
maximal normal subgroup of	#72235632, #72235672, #72235683

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