#6868101

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	#31063	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1584 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)(17,18,23,26,22,28,21,27)(19,24,30,32,31,29,20,25);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,31)(18,24,28,29)(20,23,30,21)(25,26,32,27);(10,14)(15,16)(17,18,22,28)(19,29,31,24)(20,25,30,32)(21,26,23,27);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20,22,30)(18,25,28,32)(19,21,31,23)(24,27,29,26);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,26,28,27)(19,30,31,20)(24,25,29,32)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=h,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,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, 6775297);