#6820341

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#31208	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1421 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6727537);