#453950

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2191 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)(5,6)(7,8)(9,10)(11,16)(12,17)(13,19)(14,29)(15,20)(18,25)(21,30)(22,34)(23,31)(24,33)(26,36)(27,32)(28,35)(37,40)(38,39);(6,8)(9,11,15,24)(10,16,20,33)(12,22,26,37)(13,27,14,23)(17,34,36,40)(18,30,35,39)(19,32,29,31)(21,28,38,25);(2,4)(5,6)(7,8)(9,12)(10,17)(11,21)(13,28)(14,25)(15,26)(16,30)(18,29)(19,35)(20,36)(22,27)(23,37)(24,38)(31,40)(32,34)(33,39);(6,8)(9,12)(10,18)(11,22)(13,25)(14,28)(15,26)(16,30)(17,29)(19,36)(20,35)(21,27)(23,38)(24,37)(31,34)(32,40)(33,39)

Polycyclic presentation:

2^9|b^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361146);