#6683769

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2262 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=fi,e^2=i,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,h^c=hi,e^d=ei,h^e=hi

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, 6590965);