#470834

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
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 ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

778 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=eh,c^2=fi,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de

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, 378030);