#501045

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3050	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1178 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)(9,10,15,28,26,35,25,34,14,19,13,18,12,17,30,40)(11,20,37,32,21,38,27,16,24,39,22,31,23,36,29,33);(2,4)(5,6)(7,8);(6,8)(9,11)(10,16)(12,21)(13,27)(14,24)(15,29)(17,31)(18,36)(19,33)(20,34)(22,25)(23,26)(28,38)(30,37)(32,35)(39,40);(2,4)(9,12,14,26)(10,17,19,35)(11,21,24,23)(13,25,15,30)(16,31,33,32)(18,34,28,40)(20,38,39,36)(22,29,37,27)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408241);