#364366

order	512
nilpotency class	3
isoclinism	isoclinic class of #189517
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

1074 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)(9,10,15,20,16,21,14,32)(11,22,25,19,13,31,24,17)(12,26,28,18,30,36,29,35)(23,34,38,37,27,33,39,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,25,16,13,14,24)(10,17,32,31,21,19,20,22)(12,27,29,38,30,23,28,39)(18,34,36,37,35,33,26,40);(2,6)(3,8)(9,12)(10,18,21,35)(11,23)(13,27)(14,28)(15,29)(16,30)(17,33,19,34)(20,26,32,36)(22,37,31,40)(24,38)(25,39);(2,6)(5,7)(9,13,15,24,16,11,14,25)(10,19,32,22,21,17,20,31)(12,27,29,38,30,23,28,39)(18,34,36,37,35,33,26,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271562);