#6816722

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1649 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,26,28)(23,24,30,31)(25,27,34,35)(29,32,36,33);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23,26,30)(22,24,28,31)(25,33,34,32)(27,29,35,36);(1,2,3,4)(21,24,22,23,26,31,28,30)(25,32,27,29,34,33,35,36);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(25,34)(27,35)(29,36)(32,33);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,25,28,35,26,34,22,27)(23,29,24,32,30,36,31,33)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6723918);