#449444

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
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

850 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)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,20)(19,23,24,22)(25,26)(27,34)(28,31)(29,32)(30,35)(33,37)(36,38)(39,40);(1,2,4,5)(3,7,8,6)(9,11,12,15)(10,13,14,16)(17,19,21,24)(18,22,20,23)(25,27,28,35)(26,30,31,34)(29,36,37,40)(32,39,33,38);(10,14)(13,16)(17,20,21,18)(19,22,24,23)(26,31)(27,35)(32,33)(36,40);(1,3,4,8)(2,6,5,7)(19,24)(22,23)(26,32)(27,35)(29,37)(30,38)(31,33)(34,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=efi,c^2=gi,d^2=eg,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, 356640);