#201919

order	512
nilpotency class	3
isoclinism	isoclinic class of #201918
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1922 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)(3,7,8,5)(4,6)(9,10)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,27)(22,36)(23,32)(25,31)(28,33)(29,34)(30,39)(35,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11,13,24,15,25,14,23)(10,16,19,21,20,32,18,31)(12,22,27,38,28,39,29,37)(17,30,26,35,34,36,33,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,33)(19,26)(20,34)(21,35)(23,37)(24,38)(25,39)(31,40)(32,36);(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,28,29)(16,21,32,31)(17,33,34,26)(22,37,39,38)(30,35,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,f^b=fi,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109115);