#6568978

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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

1283 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=fi,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^d=ehi

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, 6476174);