#189293

order	512
nilpotency class	3
isoclinism	isoclinic class of #189293
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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

1126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,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, 96489);