#6909018

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1535 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)(17,18,23,25)(19,22,27,21)(20,24,29,32)(26,28,31,30);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,25,23,27,21,18)(20,26,28,32,29,31,30,24);(2,6)(5,7)(11,16)(13,15)(17,20)(18,24)(19,26)(21,30)(22,28)(23,29)(25,32)(27,31);(10,14)(13,15)(17,21,23,22)(18,19,25,27)(20,28,29,30)(24,31,32,26)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6816214);