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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4035 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(8,13)(10,15)(17,18,23,28)(19,22,27,24)(20,26,21,25)(29,32,30,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,19,23,27)(18,24,28,22)(20,29,21,30)(25,32,26,31)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,20,23,21)(18,25,28,26)(19,29,27,30)(22,31,24,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,19)(18,24)(20,29)(21,30)(22,28)(23,27)(25,32)(26,31)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,3,5,9)(2,7,6,12)(4,10,11,15)(8,14,13,16)(17,21,23,20)(18,26,28,25)(19,30,27,29)(22,32,24,31);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,22,23,24)(18,27,28,19)(20,31,21,32)(25,30,26,29)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8332853);