#364365

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2280 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1058 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)(3,7)(4,6)(5,8)(9,10)(11,18)(12,26,30,36)(13,31)(14,20)(15,21)(16,24)(17,29,37,38)(19,22)(23,32,27,34)(25,33)(28,35,39,40);(1,3)(2,5)(4,8)(6,7)(9,11,22,13,15,25,14,24)(10,16,20,33,21,31,19,18)(12,27,38,39,30,23,29,28)(17,34,26,35,37,32,36,40);(2,6)(3,8)(9,12,15,30)(10,17)(11,23,25,27)(13,28,24,39)(14,29,22,38)(16,32)(18,35)(19,36)(20,26)(21,37)(31,34)(33,40);(2,6)(5,7)(9,13,22,25,15,24,14,11)(10,18,20,16,21,33,19,31)(12,28,38,27,30,39,29,23)(17,35,26,32,37,40,36,34)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=ef,d^2=eg,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271561);