#451775

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358971);