#363489

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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×#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

1314 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)(11,21)(12,24,27,17)(13,16)(14,29)(15,20)(18,23)(19,30)(22,33,36,35)(25,37,39,31)(26,34,40,38)(28,32);(1,3,4,8)(2,5,6,7)(9,11,14,13,15,23,30,28)(10,16,19,21,20,32,29,18)(12,22,26,25,27,36,40,39)(17,31,34,35,24,37,38,33);(3,8)(5,7)(9,12)(10,17)(11,22)(13,25)(14,26)(15,27)(16,31)(18,33)(19,34)(20,24)(21,35)(23,36)(28,39)(29,38)(30,40)(32,37);(2,6)(5,7)(9,13,14,23,15,28,30,11)(10,18,19,16,20,21,29,32)(12,25,26,36,27,39,40,22)(17,33,34,31,24,35,38,37)

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,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, 270685);