#454267

order	512
nilpotency class	3
isoclinism	isoclinic class of #454021
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	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1273 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)(9,10,13,18)(11,21,23,16)(12,25,26,35)(14,20,15,19)(17,29,33,27)(22,36,37,40)(24,32,28,34)(30,38,31,39);(1,3,4,8)(2,5,6,7)(9,11,13,23)(10,16,18,21)(12,22,26,37)(14,28,15,24)(17,31,33,30)(19,34,20,32)(25,40,35,36)(27,39,29,38);(1,2)(3,5)(4,6)(7,8)(9,12,13,26)(10,17,18,33)(11,22,23,37)(14,29,15,27)(16,30,21,31)(19,35,20,25)(24,39,28,38)(32,36,34,40);(2,6)(5,7)(9,12)(10,17)(11,22)(13,26)(14,27)(15,29)(16,31)(18,33)(19,25)(20,35)(21,30)(23,37)(24,38)(28,39)(32,36)(34,40)

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^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, 361463);