#6526543

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1330 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)(26,32)(27,33)(29,30)(34,35)(36,37)(39,40);(2,5)(6,8)(9,11,13,17)(10,14,15,21)(12,18,19,23)(16,20,22,24)(25,26,30,35,31,34,29,32)(27,28,33,40,37,38,36,39);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(12,19)(20,24)(25,27,29,36,31,37,30,33)(26,28,32,39,34,38,35,40);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(25,28,31,38)(26,33,34,36)(27,35,37,32)(29,39,30,40);(11,17)(14,21)(18,23)(20,24)(26,34)(28,38)(32,35)(39,40)

Polycyclic presentation:

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

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, 6433739);