#501666

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×#2218	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)(9,10,15,27,25,34,26,17,12,18,13,19,14,20,29,33)(11,21,28,31,38,32,39,35,22,16,23,36,24,37,40,30);(1,2,4,3)(5,6)(7,8)(9,11,12,22)(10,16,18,21)(13,23,15,28)(14,24,25,38)(17,30,33,35)(19,31,27,36)(20,32,34,37)(26,39,29,40);(6,8)(10,17)(13,26)(14,25)(15,29)(16,30)(18,33)(19,20)(21,35)(23,39)(24,38)(27,34)(28,40)(31,32)(36,37);(1,3,4,2)(10,18)(11,22)(17,33)(19,27)(20,34)(23,28)(24,38)(39,40)

Polycyclic presentation:

2^9|a^2=fi,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, 408862);