#6858622

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#2011	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1518 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)(9,10,13,12)(11,14,16,15)(17,18)(19,27)(20,30)(21,26)(22,24)(23,29)(25,31)(28,32);(1,2,4,5)(3,6,7,8)(9,11,13,16)(10,14,12,15)(18,24)(19,28)(25,30)(26,29);(2,5)(6,8)(9,10,13,12)(11,15,16,14)(17,19,23,30,22,28,21,25)(18,20,26,27,24,31,29,32);(1,3,4,7)(2,6,5,8)(17,20,22,31)(18,25,24,30)(19,29,28,26)(21,32,23,27);(9,12,13,10)(11,14,16,15)(17,21,22,23)(18,26,24,29)(19,25,28,30)(20,27,31,32)

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=fgh,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,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, 6765818);