#363837

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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2285 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

768 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)(3,7)(4,6)(5,8)(9,10)(11,18)(12,25,29,36)(13,30)(14,20)(15,21)(16,23)(17,28,37,38)(19,31)(22,32,26,34)(24,33)(27,35,39,40);(1,3,4,8)(2,5,6,7)(9,11,14,23,15,24,31,13)(10,16,19,18,21,30,20,33)(12,26,28,27,29,22,38,39)(17,34,36,40,37,32,25,35);(2,6)(3,8)(9,12,15,29)(10,17)(11,22,24,26)(13,27,23,39)(14,28,31,38)(16,32)(18,35)(19,36)(20,25)(21,37)(30,34)(33,40);(2,6)(5,7)(9,13,31,24,15,23,14,11)(10,18,20,16,21,33,19,30)(12,27,38,26,29,39,28,22)(17,35,25,32,37,40,36,34)

Polycyclic presentation:

2^9|a^2=h,b^2=e,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,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, 271033);