#6550574

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1334 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)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,31,32,30,34,27,33)(28,36,35,29,38,39,37,40);(2,6)(7,8)(10,14)(11,17)(16,21)(18,23)(25,27,30,31)(26,32,34,33)(28,37,38,35)(29,39,40,36);(1,3,5,4)(2,7,6,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28)(26,29)(27,35)(30,38)(31,37)(32,36)(33,39)(34,40);(1,4,5,3)(2,8,6,7)(12,19)(16,21)(18,23)(22,24)(25,29)(26,28)(27,36)(30,40)(31,39)(32,35)(33,37)(34,38);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,27,30,31)(26,33,34,32)(28,35,38,37)(29,39,40,36)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6457770);