#363488

order	512
nilpotency class	3
isoclinism	isoclinic class of #189517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#423 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1336 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)(12,24,28,17)(13,29)(14,31)(15,20)(16,22)(19,30)(21,37,25,34)(23,36)(26,32,39,33)(27,35,38,40);(1,3,4,8)(2,5,6,7)(9,11,14,22,15,23,30,13)(10,16,19,18,20,29,31,36)(12,25,27,26,28,21,38,39)(17,33,35,37,24,32,40,34);(3,8)(5,7)(9,12)(10,17)(11,21)(13,26)(14,27)(15,28)(16,32)(18,34)(19,35)(20,24)(22,39)(23,25)(29,33)(30,38)(31,40)(36,37);(2,6)(5,7)(9,13,30,23,15,22,14,11)(10,18,31,16,20,36,19,29)(12,26,38,25,28,39,27,21)(17,34,40,32,24,37,35,33)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,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, 270684);