#6769722

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24439	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1512 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6676918);