#6513605

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500489
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1923 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6420801);