#6494252

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473174
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

1750 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,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, 6401448);