#6868156

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1519 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgi,d^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6775352);