#473372

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
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

830 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efghi,c^2=e,d^2=efi,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380568);