#6683891

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1725 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)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,24,26)(23,28,31,27)(25,29,32,34)(30,35,33,36);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(22,26)(23,30)(25,32)(27,35)(28,36)(31,33);(1,2,3,4)(21,23,24,31)(22,27,26,28)(25,33,32,30)(29,35,34,36);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(22,26)(27,28)(29,34)(35,36);(7,15)(10,13)(14,20)(18,19)(21,23,24,31)(22,28,26,27)(25,33,32,30)(29,36,34,35)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,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, 6591087);