#6693134

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1434 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)(17,18)(19,25)(20,21)(22,24)(23,28)(26,31)(27,32)(29,30);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,19,21,28)(18,23,20,25)(22,31,30,27)(24,32,29,26);(9,12,13,11)(10,14,15,16)(17,20,21,18)(19,26,28,32)(22,24,30,29)(23,31,25,27);(1,3,4,7)(2,6,5,8)(18,20)(23,25)(24,29)(26,32);(2,5)(6,8)(19,27)(22,30)(23,26)(24,29)(25,32)(28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6600330);