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 | |
3896 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)(3,7)(4,8)(5,14)(6,10)(9,15)(11,13)(12,16)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32)(33,34,39,40)(35,36,43,48)(37,41,38,42)(44,46,45,47);(1,3,6,11)(2,7,10,13)(4,5,12,15)(8,14,16,9)(17,18,23,20)(19,24,21,26)(22,25,30,28)(27,31,29,32)(34,40)(35,43)(41,42)(44,45);(3,11)(4,12)(7,13)(8,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35)(34,36)(37,44)(38,45)(39,43)(40,48)(41,46)(42,47);(1,4)(2,8)(3,5)(6,12)(7,14)(9,13)(10,16)(11,15)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,36)(34,35)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45);(1,5)(2,9)(3,12)(4,11)(6,15)(7,8)(10,14)(13,16)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,37,39,38)(34,41,40,42)(35,44,43,45)(36,46,48,47);(1,2)(3,13)(4,8)(5,9)(6,10)(7,11)(12,16)(14,15)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,38,39,37)(34,42,40,41)(35,45,43,44)(36,47,48,46)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=h,e^2=gh,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=dghi,e^d=egi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9287089);