#503104

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2184	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1363 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:

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

Polycyclic presentation:

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

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, 410300);