#6568767

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1459 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=fi,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^d=eh

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, 6475963);