#435526

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

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

964 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)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,21,23,24)(25,26)(27,32)(28,35)(29,31)(30,37)(33,34)(36,40)(38,39);(1,2,4,5)(3,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,21)(20,23)(22,24)(26,31)(28,36)(32,37)(34,38);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(19,23)(21,24)(25,27,29,30)(26,32,31,37)(28,34,36,38)(33,40,39,35);(2,5)(6,8)(11,16)(13,15)(25,28,29,36)(26,33,31,39)(27,34,30,38)(32,40,37,35)

Polycyclic presentation:

2^9|a^2=fg,b^2=efi,c^2=ef,d^2=i,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, 342722);