#6463752

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442590
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

1843 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:

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

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,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, 6370948);