#6346762

order	512
nilpotency class	3
isoclinism	isoclinic class of #6346762
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3866 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253958);