#501608

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

712 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:

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

Polycyclic presentation:

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

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, 408804);