#72309190

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

5791 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=e,e^2=h,f^2=j,g^2=j,b^a=bf,c^b=cij,d^b=dhj,e^a=ej,f^a=fi,g^b=gij

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309195, #72309367
maximal normal subgroup of	#72309195, #72309367

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