#498850

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#508	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1558 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)(9,10)(11,20)(12,22)(13,17)(14,28)(15,19)(16,33)(18,32)(21,30)(23,37)(24,40)(25,38)(26,39)(27,35)(29,34)(31,36);(2,4)(5,6)(7,8)(9,11,15,23)(10,12,19,26)(13,24,16,21)(14,29,31,38)(17,34,33,25)(18,30,35,40)(20,36,37,28)(22,27,39,32);(6,8)(9,12)(10,11)(13,25)(14,30)(15,26)(16,34)(17,21)(18,29)(19,23)(20,32)(22,28)(24,33)(27,37)(31,40)(35,38)(36,39);(9,13,15,16)(10,17,19,33)(11,21,23,24)(12,25,26,34)(14,27,31,32)(18,28,35,36)(20,30,37,40)(22,38,39,29)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,c^b=cg,d^b=di,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406046);