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 | |
4228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,10,8)(4,13,15,5)(11,16,14,12)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,28)(29,32)(33,34,39,44)(35,42,37,40)(36,41,47,45)(38,46,48,43);(2,7)(5,13)(8,9)(12,16)(17,19)(18,23)(20,21)(22,27)(24,30)(25,32)(26,28)(29,31)(35,37)(38,48)(40,42)(43,46);(3,10)(8,9)(11,14)(12,16)(19,27)(20,29)(23,28)(24,32)(33,35,39,37)(34,40,44,42)(36,46,47,43)(38,45,48,41);(1,3)(2,8)(4,14)(5,12)(6,10)(7,9)(11,15)(13,16)(17,20,22,29)(18,24,26,32)(19,21,27,31)(23,30,28,25)(33,36)(34,41)(35,43)(37,46)(38,42)(39,47)(40,48)(44,45);(1,4)(2,5)(3,11)(6,15)(7,13)(8,16)(9,12)(10,14)(17,21)(18,25)(19,20)(22,31)(23,32)(24,28)(26,30)(27,29)(33,37,39,35)(34,42,44,40)(36,43,47,46)(38,41,48,45);(1,5)(2,4)(3,12)(6,13)(7,15)(8,14)(9,11)(10,16)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31)(33,38,39,48)(34,43,44,46)(35,45,37,41)(36,42,47,40)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=hi,e^2=h,f^2=h,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=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8746326);