#499325

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

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^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, 406521);