#6516933

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500226
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 ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3286 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6424129);