#386946

order	512
nilpotency class	3
isoclinism	isoclinic class of #219696
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
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

898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 294142);