#7611883

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563333
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

1922 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)(3,5)(4,6)(7,8)(9,10)(11,23)(12,27)(13,32)(14,37)(15,28)(16,22)(17,21)(18,19)(20,35)(24,38)(25,30)(26,31)(29,33)(34,39)(36,40);(1,3)(2,5)(4,7)(6,8)(9,11,15,19,16,26,17,27)(10,18,21,23,22,12,28,31)(13,33,14,25,35,39,36,24)(20,38,37,34,32,30,40,29);(2,6)(3,7)(9,12,16,18)(10,19,22,27)(11,21,26,28)(13,34,35,29)(14,30,36,38)(15,31,17,23)(20,39,32,33)(24,40,25,37);(2,6)(5,8)(9,13,17,36,16,35,15,14)(10,20,21,37,22,32,28,40)(11,24,19,33,26,25,27,39)(12,29,23,30,18,34,31,38);(9,14,17,13,16,36,15,35)(10,20,28,40,22,32,21,37)(11,25,27,33,26,24,19,39)(12,30,23,34,18,38,31,29)

Polycyclic presentation:

2^9|b^2=f,c^2=i,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,d^c=di,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, 7519079);