#198133

order	512
nilpotency class	3
isoclinism	isoclinic class of #198126
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×#3005	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2069 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)(3,5,7,8)(4,6)(9,10)(11,21)(12,25,28,34)(13,18)(14,19)(15,20)(16,23)(17,26,33,27)(22,36,39,30)(24,31)(29,32)(35,38,40,37);(1,3)(2,5,6,8)(4,7)(9,11,13,23,15,24,14,29)(10,16,19,21,20,32,18,31)(12,22,26,37,28,39,27,38)(17,30,34,35,33,36,25,40);(9,12)(10,17)(11,22)(13,26)(14,27)(15,28)(16,30)(18,25)(19,34)(20,33)(21,35)(23,37)(24,39)(29,38)(31,40)(32,36);(2,6)(5,8)(9,13,15,14)(10,18,20,19)(11,23,24,29)(12,26,28,27)(16,31,32,21)(17,25,33,34)(22,37,39,38)(30,40,36,35)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105329);