#474248

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2572
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

671 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)(3,6)(4,5)(7,8)(9,10,13,17,14,18,28,19)(11,15,22,32,21,31,39,30)(12,23,25,34,27,16,26,35)(20,36,24,40,37,29,38,33);(2,5)(6,8)(9,11,14,21)(10,15,18,31)(12,24)(13,22,28,39)(16,33)(17,32,19,30)(20,26)(23,40)(25,37)(27,38)(29,34)(35,36);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,20)(13,26)(14,27)(15,29)(17,34)(18,23)(19,35)(21,37)(22,38)(24,39)(25,28)(30,33)(31,36)(32,40);(3,7)(6,8)(10,17,18,19)(11,21)(15,30,31,32)(16,34,23,35)(20,37)(22,39)(24,38)(29,33,36,40)

Polycyclic presentation:

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

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, 381444);