#497440

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1006 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)(5,6,7,8)(9,10)(11,19)(12,18)(13,28)(15,16)(17,22)(20,34)(21,35)(23,37)(24,39)(26,27)(29,31)(32,36)(33,38);(9,11,16,21)(10,17,15,31)(12,23,27,20)(13,29,28,22)(14,19,30,35)(18,33,26,32)(24,36,39,38)(25,37,40,34);(2,4)(9,12,16,27)(10,18,15,26)(11,20,21,23)(13,24,28,39)(14,25,30,40)(17,32,31,33)(19,34,35,37)(22,36,29,38);(9,13)(10,14)(11,21)(12,24)(15,30)(16,28)(18,25)(19,29)(20,23)(22,35)(26,40)(27,39)(34,38)(36,37)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,c^b=ci,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, 404636);