#381258

order	512
nilpotency class	3
isoclinism	isoclinic class of #214688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1255 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)(9,10,15,22,16,20,14,21)(11,23,26,32,12,17,25,18)(13,19,28,36,29,35,30,37)(24,38,39,34,27,31,40,33);(2,5)(6,8)(9,11,16,12)(10,17,20,23)(13,24,29,27)(14,26,15,25)(18,22,32,21)(19,31,35,38)(28,40,30,39)(33,36,34,37);(3,7)(6,8)(9,12,16,11)(10,18,20,32)(13,27,29,24)(14,25,15,26)(17,21,23,22)(19,34,35,33)(28,39,30,40)(31,36,38,37);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,24)(12,27)(14,30)(15,28)(16,29)(17,31)(18,33)(20,35)(21,37)(22,36)(23,38)(25,40)(26,39)(32,34)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,d^c=de,f^c=fi,h^c=hi,f^d=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, 288454);