#6498985

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470922
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1234 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gh,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6406181);