#440966

order	512
nilpotency class	3
isoclinism	isoclinic class of #419454
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1065 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348162);