#6513283

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500006
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

2986 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)(11,19,22,28)(16,30,20,32)(17,29)(18,27)(21,35,23,36)(31,39,33,40)(34,38);(1,2)(3,6)(4,5)(7,8)(9,10,13,15)(11,20,22,16)(12,17,24,29)(14,27,26,18)(19,32,28,30)(21,33,23,31)(25,38,37,34)(35,40,36,39);(2,5)(3,7)(9,11)(10,16)(12,23)(13,22)(14,28)(15,20)(17,33)(18,30)(19,26)(21,24)(25,35)(27,32)(29,31)(34,40)(36,37)(38,39);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,24)(14,25)(15,29)(16,31)(18,34)(19,35)(20,33)(22,23)(26,37)(27,38)(28,36)(30,39)(32,40);(2,5)(6,8)(10,15)(12,24)(16,20)(18,27)(21,23)(25,37)(30,32)(35,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=hi,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,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, 6420479);