#201255

order	512
nilpotency class	3
isoclinism	isoclinic class of #201112
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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1245 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,g^c=ghi

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, 108451);