#6684109

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1666 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6591305);