#201627

order	512
nilpotency class	3
isoclinism	isoclinic class of #201404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1418 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,f^b=fi,e^c=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108823);