#499326

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ehi,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^c=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, 406522);