#7611888

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563327
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

1964 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=h,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,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, 7519084);