#501662

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
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	C₂×#2215	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

715 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408858);