#6854451

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1340 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=ghi,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6761647);