#72309128

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	C₂⁴	C₂²×C₄²
Frattini, cosocle	C₂²×C₄²	C₂⁴
socle	C₂³	C₂²×#95
prime radical	C₂⁶	C₄²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

5669 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)(17,18,20,24,21,23,28,31)(19,22,26,30,25,29,27,32);(2,5)(6,8)(11,16)(13,15);(2,5)(6,8)(19,25)(22,29)(26,27)(30,32);(1,2,3,6,4,5,7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,22,23,29)(20,27)(21,25)(24,32,31,30)(26,28)

Polycyclic presentation:

2^10|a^2=g,d^2=e,e^2=h,g^2=j,d^a=df,d^b=dhij,d^c=dhj,e^a=ej,f^a=fi,f^d=fj,g^d=gi

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309133, #72309366
maximal normal subgroup of	#72309133, #72309366

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