#7611866

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

1961 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)(9,10,17,20)(11,15,23,16)(12,18,28,35)(13,19,31,29)(14,22,30,39)(21,26,38,27)(24,33,40,34)(25,37,32,36);(1,3)(2,5)(4,7)(6,8)(9,11,15,20,17,23,16,10)(12,21,26,35,28,38,27,18)(13,29,30,39,31,19,14,22)(24,34,25,37,40,33,32,36);(2,6)(3,7)(9,12)(10,18)(11,21)(13,24)(14,32)(15,26)(16,27)(17,28)(19,33)(20,35)(22,36)(23,38)(25,30)(29,34)(31,40)(37,39);(2,6)(5,8)(9,13,16,14,17,31,15,30)(10,19,11,22,20,29,23,39)(12,24,27,32,28,40,26,25)(18,33,21,36,35,34,38,37);(9,14,15,13,17,30,16,31)(10,19,11,22,20,29,23,39)(12,25,26,40,28,32,27,24)(18,34,21,37,35,33,38,36)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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,e^c=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, 7519062);