#501618

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×#2206	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

739 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)(11,19)(12,15)(13,17)(14,18)(16,21)(20,30)(22,36)(23,28)(24,29)(25,35)(26,40)(27,32)(31,38)(33,37)(34,39);(1,2,4,3)(10,15)(11,20)(13,25)(14,24)(16,30)(17,33)(19,31)(21,38)(22,39)(23,37)(26,27)(28,35)(32,34)(36,40);(6,8)(9,11,12,21)(10,16,15,19)(13,26,23,39)(14,27,24,22)(17,34,28,40)(18,36,29,32)(20,37,38,25)(30,35,31,33);(1,3,4,2)(10,15)(11,21)(17,28)(18,29)(20,38)(22,27)(26,39)(33,35)

Polycyclic presentation:

2^9|a^2=fi,b^2=ei,c^2=i,d^2=ei,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, 408814);