#454290

order	512
nilpotency class	3
isoclinism	isoclinic class of #454029
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1302 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361486);