#117492

order	512
nilpotency class	3
isoclinism	isoclinic class of #117489
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

659 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=hi,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,e^b=eh,f^b=fhi,d^c=dh,e^c=ehi,f^c=fi,g^c=gh

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, 24688);