#201112

order	512
nilpotency class	3
isoclinism	isoclinic class of #201112
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	#35149	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2109 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)(11,19)(12,23,25,16)(13,26)(14,18)(15,22)(17,28)(20,33,35,40)(21,34)(24,31,27,32)(29,36,39,38)(30,37);(1,3)(2,5,6,8)(4,7)(9,11,13,22,14,21,28,37)(10,15,17,19,18,30,26,34)(12,20,27,38,25,35,24,36)(16,29,32,33,23,39,31,40);(9,12)(10,16)(11,20)(13,27)(14,25)(15,29)(17,32)(18,23)(19,33)(21,35)(22,38)(24,28)(26,31)(30,39)(34,40)(36,37);(2,6)(3,7)(11,21)(15,30)(19,34)(20,35)(22,37)(29,39)(33,40)(36,38)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=hi,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, 108308);