#6512745

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499849
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3065 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)(11,21)(12,25)(13,18)(14,19)(15,20)(16,23)(17,28)(22,32)(24,33)(26,35)(27,31)(29,36)(30,34)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,26)(13,30)(14,23)(15,24)(17,35)(18,21)(19,33)(20,34)(22,29)(25,37)(27,32)(28,38)(31,39)(36,40);(2,5)(3,7)(9,12)(10,17)(11,22)(13,28)(14,31)(15,29)(16,32)(18,36)(19,25)(20,27)(21,37)(23,38)(24,26)(30,39)(33,40)(34,35);(1,3,4,7)(2,6,5,8)(9,10)(11,16)(12,27)(13,19)(14,18)(15,20)(17,29)(21,23)(22,35)(24,34)(25,31)(26,32)(28,36)(30,33)(37,39)(38,40);(2,5)(6,8)(9,13,15,14)(10,18,20,19)(11,23,24,30)(12,28,29,31)(16,33,34,21)(17,36,27,25)(22,38,26,39)(32,40,35,37)

Polycyclic presentation:

2^9|d^2=hi,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,f^b=fi,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, 6419941);