#7611860

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

1925 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)(11,14)(12,19)(13,20)(15,31)(16,21)(17,22)(18,24)(23,27)(25,33)(26,34)(28,40)(29,36)(30,37)(32,35)(38,39);(1,3)(2,5)(4,7)(6,8)(9,11,15,24,17,25,16,20)(10,18,21,14,22,13,31,33)(12,23,28,34,30,38,29,35)(19,26,36,27,37,32,40,39);(2,6)(3,7)(9,12,17,30)(10,19,22,37)(11,23,25,38)(13,32,18,26)(14,27,33,39)(15,28,16,29)(20,35,24,34)(21,36,31,40);(2,6)(5,8)(9,13,16,14,17,18,15,33)(10,20,21,11,22,24,31,25)(12,26,29,39,30,32,28,27)(19,34,36,38,37,35,40,23);(9,14,15,13,17,33,16,18)(10,20,21,11,22,24,31,25)(12,27,28,32,30,39,29,26)(19,35,36,23,37,34,40,38)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=h,e^2=f,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

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, 7519056);