#419027

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
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
prime radical	C₂×#3012	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2334 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=dei,f^c=fi,e^d=ei,h^d=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, 326223);