#6499086

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1305 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,19)(13,15)(16,20)(17,21)(18,24)(22,23)(25,26,31,36)(27,37,38,39)(28,33,32,40)(29,35,30,34);(2,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,22)(19,24)(20,23)(27,38)(28,32)(33,40)(37,39);(1,3,5,4)(2,7,6,8)(10,15)(11,17)(16,19)(18,23)(25,27,30,33,31,38,29,40)(26,32,35,37,36,28,34,39);(1,4,5,3)(2,8,6,7)(11,17)(14,21)(18,23)(22,24)(25,28)(26,33)(27,34)(29,37)(30,39)(31,32)(35,38)(36,40);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,22,21,24)(25,29,31,30)(26,34,36,35)(27,33,38,40)(28,39,32,37)

Polycyclic presentation:

2^9|a^2=i,c^2=fghi,d^2=f,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^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6406282);