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 | |
4056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,9,11)(5,12,13,8)(10,16,15,14)(17,18,22,26)(19,24,29,23)(20,30,25,21)(27,28,31,32)(33,34,39,37)(35,40,45,43)(36,41,47,38)(42,44,46,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19,22,29)(18,23,26,24)(20,27,25,31)(21,28,30,32)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47);(2,4)(3,9)(5,13)(14,16)(17,19,22,29)(18,23,26,24)(20,31,25,27)(21,32,30,28)(33,36,39,47)(34,41,37,38)(35,42,45,46)(40,44,43,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20,22,25)(18,21,26,30)(19,27,29,31)(23,28,24,32)(33,37,39,34)(35,43,45,40)(36,41,47,38)(42,44,46,48);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,19)(18,24)(20,31)(21,28)(22,29)(23,26)(25,27)(30,32)(33,38,39,41)(34,36,37,47)(35,44,45,48)(40,46,43,42);(1,5,6,13)(2,8,4,12)(3,10,9,15)(7,14,11,16)(17,21)(18,25)(19,28)(20,26)(22,30)(23,31)(24,27)(29,32)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=hi,d^2=hi,e^2=i,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, 8922627);