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 | |
4097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(6,10)(11,15)(14,16)(17,18)(19,22)(20,28)(21,25)(23,27)(24,30)(26,32)(29,31)(33,34)(35,44)(36,47)(37,38)(39,40)(41,45)(42,46)(43,48);(1,2)(3,6)(4,12)(5,8)(7,13)(9,10)(11,16)(14,15)(17,19,21,27)(18,22,25,23)(20,29,30,26)(24,32,28,31)(34,40)(35,45)(37,43)(42,47);(1,2,5,8)(3,6,9,10)(4,7,13,12)(11,14,15,16)(17,18,21,25)(19,23,27,22)(20,24,30,28)(26,32,29,31)(33,35)(34,41)(36,43)(37,46)(38,47)(39,45)(40,44)(42,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,13)(14,15)(17,18)(19,22)(20,24)(21,25)(23,27)(26,31)(28,30)(29,32)(33,36,39,46)(34,42,40,47)(35,43,45,37)(38,41,48,44);(1,3)(2,6)(4,11)(5,9)(7,14)(8,10)(12,16)(13,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,32)(28,31)(29,30)(33,37,39,43)(34,38,40,48)(35,46,45,36)(41,47,44,42);(1,4)(2,7)(3,11)(5,13)(6,14)(8,12)(9,15)(10,16)(17,20,21,30)(18,24,25,28)(19,26,27,29)(22,31,23,32)(33,38)(34,43)(35,47)(36,41)(37,40)(39,48)(42,45)(44,46)Polycyclic presentation:
2^9|b^2=gh,c^2=gi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8738315);