#6464100

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443050
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 ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1732 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:

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

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^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, 6371296);