#469195

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

880 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=efg,c^2=egi,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376391);