Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #47575 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3975 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,5,9,7)(4,8,12,16)(11,14,13,15)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31)(33,34,38,37)(35,39,44,42)(36,40,45,46)(41,47,43,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,15)(12,16)(13,14)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47);(3,9)(4,12)(5,7)(8,16)(17,18,22,26)(19,24,20,23)(21,29,30,25)(27,31,28,32)(33,36,38,45)(34,40,37,46)(35,41,44,43)(39,47,42,48);(1,3,6,9)(2,7,10,5)(4,13,12,11)(8,14,16,15)(17,20,22,19)(18,24,26,23)(21,28,30,27)(25,32,29,31)(33,37,38,34)(35,42,44,39)(36,40,45,46)(41,47,43,48);(1,4,6,12)(2,8,10,16)(3,11,9,13)(5,14,7,15)(17,18)(19,24)(20,23)(21,25)(22,26)(27,32)(28,31)(29,30)(33,36)(34,40)(35,43)(37,46)(38,45)(39,48)(41,44)(42,47);(1,5,6,7)(2,9,10,3)(4,14,12,15)(8,13,16,11)(17,21,22,30)(18,25,26,29)(19,27,20,28)(23,31,24,32)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=i,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8900441);