#381257

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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1311 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)(9,10,14,17,15,19,13,20)(11,21,24,31,25,16,23,29)(12,18,26,32,27,34,28,33)(22,35,36,40,37,30,38,39);(2,5)(6,8)(9,11,15,25)(10,16,19,21)(12,22,27,37)(13,24,14,23)(17,31,20,29)(18,30,34,35)(26,38,28,36)(32,40,33,39);(3,7)(6,8)(10,17)(13,14)(16,29)(18,33)(19,20)(21,31)(23,24)(26,28)(30,40)(32,34)(35,39)(36,38);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,22)(13,28)(14,26)(15,27)(16,30)(17,32)(19,34)(20,33)(21,35)(23,38)(24,36)(25,37)(29,39)(31,40)

Polycyclic presentation:

2^9|a^2=h,b^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, 288453);