#7613488

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

1876 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,23)(13,26)(14,29)(15,20)(17,22)(18,28)(19,30)(21,35)(24,31)(25,32)(27,39)(33,38)(34,40)(36,37);(2,5)(6,8)(9,11,15,22)(10,16,20,17)(12,14,25,30)(13,21,27,36)(18,33,34,31)(19,23,29,32)(24,28,38,40)(26,35,39,37);(1,3,4,7)(2,6,5,8)(9,12,14,22,15,25,30,11)(10,17,19,23,20,16,29,32)(13,24,28,36,27,38,40,21)(18,33,26,37,34,31,39,35);(9,13,15,27)(10,18)(11,21,22,36)(12,24,25,38)(14,28,30,40)(16,31)(17,33)(19,26)(20,34)(23,37)(29,39)(32,35);(2,5)(6,8)(13,27)(18,34)(21,36)(24,38)(26,39)(28,40)(31,33)(35,37)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,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, 7520684);