#6464095

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443048
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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1756 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=g,d^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6371291);