#499329

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	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

864 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,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, 406525);