#363539

order	512
nilpotency class	3
isoclinism	isoclinic class of #189517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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×#380
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#423 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1020 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,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, 270735);