#509605

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

679 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,h^a=hi,c^b=cg,d^b=de,e^b=eh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416801);