#6674424

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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×#3197	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6581620);