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 | |
4074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,10,5)(4,8,12,16)(11,15,13,14)(17,18,22,24)(19,25,26,21)(20,23,28,31)(27,32,29,30)(33,34,39,43)(35,40,38,48)(36,37,42,41)(44,46,45,47);(1,3)(2,7)(4,11)(5,9)(6,10)(8,15)(12,13)(14,16)(17,18)(19,21)(20,23)(22,24)(25,26)(27,30)(28,31)(29,32)(33,35,39,38)(34,40,43,48)(36,45,42,44)(37,47,41,46);(3,10)(4,12)(5,7)(8,16)(19,26)(20,28)(21,25)(23,31)(33,36,39,42)(34,41,43,37)(35,44,38,45)(40,47,48,46);(1,3)(2,7)(4,13)(5,9)(6,10)(8,14)(11,12)(15,16)(17,19)(18,21)(20,29)(22,26)(23,32)(24,25)(27,28)(30,31)(33,36)(34,37)(35,45)(38,44)(39,42)(40,47)(41,43)(46,48);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,16)(10,13)(17,20)(18,23)(19,27)(21,30)(22,28)(24,31)(25,32)(26,29)(33,37)(34,42)(35,46)(36,43)(38,47)(39,41)(40,45)(44,48);(1,5,6,7)(2,3,9,10)(4,14,12,15)(8,11,16,13)(17,21)(18,19)(20,30)(22,25)(23,27)(24,26)(28,32)(29,31)(33,38,39,35)(34,40,43,48)(36,44,42,45)(37,47,41,46)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,f^2=i,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, 8927265);