#502047

order	512
nilpotency class	4
isoclinism	isoclinic class of #501810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

445 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409243);