#497873

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

666 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405069);