#509293

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

553 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ei,d^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416489);