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 | |
4119 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(19,25)(23,26)(27,31)(30,32)(33,34,40,45)(35,46,37,41)(36,47,48,42)(38,43,44,39);(2,7)(4,8)(10,14)(12,15)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,35,40,37)(34,41,45,46)(36,44,48,38)(39,47,43,42);(3,9)(4,8)(5,13)(10,14)(18,22)(19,25)(20,28)(30,32)(33,36,40,48)(34,42,45,47)(35,44,37,38)(39,46,43,41);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,37,40,35)(34,41,45,46)(36,44,48,38)(39,42,43,47);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,38,40,44)(34,43,45,39)(35,36,37,48)(41,42,46,47);(1,5,6,13)(2,10,7,14)(3,11,9,16)(4,12,8,15)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,39,40,43)(34,44,45,38)(35,47,37,42)(36,41,48,46)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8964572);