#6826643

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1304 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,31)(21,29)(22,25)(23,26)(24,30)(28,32);(9,11)(10,13)(12,16)(14,15)(17,19,22,29)(18,21,25,27)(20,32,30,23)(24,28,31,26);(2,5)(6,8)(11,16)(13,15)(17,20,23,19,22,30,32,29)(18,24,26,21,25,31,28,27);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,25)(21,27)(24,31)(26,28);(17,21,32,24,22,27,23,31)(18,19,28,20,25,29,26,30)

Polycyclic presentation:

2^9|b^2=i,c^2=fhi,d^2=gi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,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, 6733839);