#474804

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

874 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:

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

Polycyclic presentation:

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

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, 382000);