#6463747

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442587
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	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1932 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=gi,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^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, 6370943);