#6755905

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1418 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)(11,19)(12,16)(13,18)(14,17)(15,21)(20,24)(22,23)(25,26,29,33)(27,35,37,31)(28,38,39,32)(30,40,36,34);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,17,19)(12,20,13,22)(16,23,18,24)(25,27)(26,31)(28,30)(29,37)(32,40)(33,35)(34,38)(36,39);(1,3,4,8)(2,5,6,7)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28)(26,31)(27,30)(29,39)(32,34)(33,35)(36,37)(38,40);(10,17)(15,19)(16,18)(23,24)(25,28)(26,32)(27,36)(29,39)(30,37)(31,34)(33,38)(35,40);(2,6)(5,7)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(27,37)(28,39)(31,35)(32,38)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fhi,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,e^c=ei,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, 6663101);