#501665

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×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

694 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)(9,10,15,26,25,34,24,17,14,20,12,18,13,19,28,33)(11,16,27,37,35,40,36,29,21,30,22,31,23,32,38,39);(1,2,4,3)(5,6)(7,8)(9,11)(10,16)(12,22)(13,23)(14,21)(15,27)(17,29)(18,31)(19,32)(20,30)(24,36)(25,35)(26,37)(28,38)(33,39)(34,40);(6,8)(10,17)(12,24)(13,25)(15,28)(16,29)(18,19)(20,33)(22,36)(23,35)(26,34)(27,38)(30,39)(31,32)(37,40);(1,3,4,2)(11,21)(16,30)(22,27)(23,35)(29,39)(31,37)(32,40)(36,38)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=e,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, 408861);