#6791741

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1431 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,29,33)(27,30,34,39)(28,36,38,32)(31,37,40,35);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,29,34)(26,30,33,39)(28,35,38,37)(31,32,40,36);(2,5)(6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,27)(26,31)(28,37)(29,34)(30,36)(32,39)(33,40)(35,38);(10,16)(14,17)(15,22)(21,23)(25,27)(26,30)(28,35)(29,34)(31,36)(32,40)(33,39)(37,38);(1,3,4,8)(2,6,5,7)(11,19)(14,17)(18,24)(21,23)(27,34)(30,39)(31,40)(35,37)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,e^2=hi,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, 6698937);