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 | |
4225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,4)(5,13)(6,8)(7,9)(10,14)(11,15)(12,16)(33,34,38,43)(35,39,46,48)(36,41,45,40)(37,42,44,47);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(17,18,22,23)(19,24,27,32)(20,25,30,29)(21,26,31,28)(33,35,38,46)(34,39,43,48)(36,44,45,37)(40,42,41,47);(2,8)(4,7)(5,14)(11,16)(18,23)(19,27)(20,30)(21,31)(33,36,38,45)(34,40,43,41)(35,37,46,44)(39,47,48,42);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,19)(18,24)(20,26)(21,29)(22,27)(23,32)(25,31)(28,30)(33,36)(34,41)(35,44)(37,46)(38,45)(39,47)(40,43)(42,48);(1,2)(3,4)(5,10)(6,8)(7,9)(11,12)(13,14)(15,16)(17,20)(18,25)(19,28)(21,24)(22,30)(23,29)(26,27)(31,32)(33,37,38,44)(34,42,43,47)(35,45,46,36)(39,40,48,41);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,21)(18,26)(19,29)(20,24)(22,31)(23,28)(25,27)(30,32)(33,34)(35,39)(36,41)(37,42)(38,43)(40,45)(44,47)(46,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=g,e^2=gh,f^2=g,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, 8884559);