#6756157

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,26,30,34)(27,31,29,38)(28,33,37,39)(32,36,40,35);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24)(25,27,30,29)(26,31,34,38)(28,35,37,36)(32,39,40,33);(1,3,4,8)(2,5,6,7)(25,28)(26,32)(27,35)(29,36)(30,37)(31,39)(33,38)(34,40);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,28)(26,33)(27,36)(29,35)(30,37)(31,40)(32,38)(34,39);(2,6)(5,7)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,27,30,29)(26,31,34,38)(28,35,37,36)(32,39,40,33)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6663353);