#72313871

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×C₂²⋊#95
commutator	C₂²×C₄	C₂²×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	C₂×#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

5191 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:

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

Polycyclic presentation:

2^10|b^2=e,c^2=i,d^2=g,f^2=j,g^2=j,b^a=bh,d^b=df,d^c=dj,e^a=ei,e^d=ej,f^a=fij,g^b=gj,h^b=hi,h^d=hij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313915
maximal normal subgroup of	#72313915

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