#6684137

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1711 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)(23,28)(24,32)(25,27)(26,30)(29,35)(31,34)(33,36);(1,2)(3,4)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,27,30)(22,26,25,28)(24,31,34,32)(29,33,36,35);(2,4)(21,24,28,35,27,34,26,33)(22,29,30,32,25,36,23,31);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,25,27,22)(23,26,30,28)(24,29,34,36)(31,35,32,33);(2,4)(7,15)(10,13)(14,20)(18,19)(21,26,27,28)(22,30,25,23)(24,33,34,35)(29,32,36,31)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=g,d^2=f,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,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, 6591333);