#6516929

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500157
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3340 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:

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

Polycyclic presentation:

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

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, 6424125);