#6526536

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1419 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)(27,34)(29,32)(33,38)(35,40)(37,39);(2,5)(6,8)(9,11,13,17)(10,14,15,21)(12,18,19,23)(16,20,22,24)(25,26,29,31,30,27,32,34)(28,33,37,38,36,35,39,40);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(12,19)(20,24)(25,27,29,34,30,26,32,31)(28,35,37,40,36,33,39,38);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(25,28,30,36)(26,33,27,35)(29,37,32,39)(31,38,34,40);(11,17)(14,21)(18,23)(20,24)(28,36)(33,35)(37,39)(38,40)

Polycyclic presentation:

2^9|a^2=fghi,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=eh,d^c=dh,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, 6433732);