#6868010

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#31057	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=fg,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6775206);