#6463745

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442584
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1875 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6370941);