#499331

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

808 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)(11,23)(12,19)(13,21)(14,20)(15,31)(16,22)(17,25)(18,26)(24,35)(27,38)(28,32)(29,36)(30,37)(33,40)(34,39);(5,6)(7,8)(9,11,14,26,15,24,30,39,16,27,29,40,12,28,13,25)(10,17,21,32,19,33,36,38,22,34,37,35,31,18,20,23);(2,4)(6,8)(9,11)(10,18)(12,24)(13,26)(14,25)(15,28)(16,27)(17,31)(19,34)(20,23)(21,35)(22,33)(29,39)(30,40)(32,37)(36,38);(9,12,16,15)(10,19,22,31)(11,24,27,28)(13,29,30,14)(17,18,34,33)(20,21,36,37)(23,35,38,32)(25,26,39,40)

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,d^2=i,e^2=hi,f^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,h^a=hi,c^b=cg,d^b=di,e^c=eh,f^c=fhi,g^c=ghi,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, 406527);