#189353

order	512
nilpotency class	3
isoclinism	isoclinic class of #189298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1053 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)(3,5)(4,6)(7,8)(9,10,14,19,15,20,13,18)(11,21,24,32,25,31,23,16)(12,26,27,17,29,33,28,34)(22,36,37,35,39,30,38,40);(9,11,13,23,15,25,14,24)(10,16,19,21,20,32,18,31)(12,22,27,37,29,39,28,38)(17,30,26,35,34,36,33,40);(2,5)(3,8)(6,7)(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,33)(19,26)(20,34)(21,35)(23,37)(24,38)(25,39)(31,40)(32,36);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,29,28)(16,31,32,21)(17,33,34,26)(22,37,39,38)(30,40,36,35)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,f^c=fhi,g^c=gh

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, 96549);