#503116

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ 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)(11,13)(12,22)(14,29)(15,21)(16,25)(17,23)(18,28)(19,31)(20,35)(24,33)(26,39)(27,32)(30,40)(34,37)(36,38);(1,2,4,3)(5,6)(7,8)(9,11,13,10)(12,19,24,16)(14,20,27,17)(15,21,28,18)(22,25,33,31)(23,32,35,29)(26,34,38,30)(36,37,39,40);(6,8)(9,12,23,37,15,26,27,31,13,24,35,40,28,38,14,25)(10,16,29,36,18,30,20,33,11,19,32,39,21,34,17,22);(1,3,4,2)(10,11)(16,19)(17,20)(18,21)(22,33)(29,32)(30,34)(36,39)

Polycyclic presentation:

2^9|b^2=e,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,d^b=di

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, 410312);