#6816088

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814811
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2062 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:

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

Polycyclic presentation:

2^9|c^2=gh,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723284);