#509603

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×#2218	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

703 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)(9,10,14,19)(11,20,21,16)(12,24,27,34)(13,15,28,31)(17,25,35,40)(18,36,29,23)(22,30,26,39)(32,37,33,38);(5,6)(7,8)(10,15)(11,21)(12,25)(13,23)(16,18)(19,31)(20,29)(22,37)(24,39)(26,38)(27,40)(28,36)(30,34)(32,33);(2,4)(6,8)(9,11,14,21)(10,16,19,20)(12,26,27,22)(13,23,28,36)(15,29,31,18)(17,33,35,32)(24,30,34,39)(25,37,40,38);(2,4)(9,12,23,38,11,22,28,40,14,27,36,37,21,26,13,25)(10,17,15,30,16,32,29,24,19,35,31,39,20,33,18,34)

Polycyclic presentation:

2^9|a^2=i,c^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, 416799);