#201590

order	512
nilpotency class	3
isoclinism	isoclinic class of #201424
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1263 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)(3,4)(5,7)(6,8)(9,10,14,19,15,18,13,29)(11,20,24,31,22,35,23,16)(12,25,27,33,28,17,26,32)(21,34,38,30,36,40,37,39);(1,3,4,2,6,7,5,8)(9,11,15,22)(10,16)(12,21,28,36)(13,23,14,24)(17,30)(18,31)(19,20)(25,39)(26,37,27,38)(29,35)(32,40)(33,34);(1,4,6,5)(2,7,8,3)(9,12,15,28)(10,17,18,25)(11,21,22,36)(13,27,14,26)(16,30,31,39)(19,33,29,32)(20,34,35,40)(23,38,24,37);(1,4,6,5)(2,3,8,7)(10,18)(11,22)(17,25)(19,29)(21,36)(23,24)(32,33)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,f^b=fi,e^c=ei,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, 108786);