#201119

order	512
nilpotency class	3
isoclinism	isoclinic class of #201112
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	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1212 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,6,10,13)(4,7)(5,8)(9,12,15,16)(11,14)(17,18)(19,28)(20,32,34,25)(21,26)(22,36)(23,27)(24,30)(29,43,46,48)(31,39)(33,42,37,41)(35,40)(38,45,47,44);(1,3)(2,6,8,13)(4,9)(5,10)(7,12,14,16)(11,15)(17,19,22,30,23,31,21,35)(18,24,26,28,27,40,36,39)(20,29,37,44,34,46,33,45)(25,38,41,43,32,47,42,48);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,12,13,16)(17,20,23,34)(18,25,27,32)(19,29,31,46)(21,33,22,37)(24,38,40,47)(26,41,36,42)(28,43,39,48)(30,44,35,45);(2,8)(3,10)(7,14)(9,15)(17,21,23,22)(18,26,27,36)(19,30,31,35)(20,33,34,37)(24,39,40,28)(25,41,32,42)(29,44,46,45)(38,48,47,43)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=hi,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, 108315);