#437558

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

902 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)(9,10,13,16)(11,14,12,15)(17,18,20,22)(19,21,23,24)(25,26)(27,34)(28,32)(29,31)(30,35)(33,37)(36,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,16,15)(17,19)(18,21)(20,23)(22,24)(25,27)(26,30)(28,35)(29,36)(31,39)(32,34)(33,38)(37,40);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,16,14)(19,23)(21,24);(2,5)(6,8)(9,12,13,11)(10,15,16,14)(26,31)(27,35)(29,37)(30,38)(32,33)(34,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 344754);