#201128

order	512
nilpotency class	3
isoclinism	isoclinic class of #201121
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108324);