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 | |
3979 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)(3,8)(4,9)(5,12)(6,7)(10,15)(11,14)(13,16)(18,20)(19,22)(25,28)(27,29)(33,34)(35,41)(36,43)(37,40)(38,39)(42,47)(44,48)(45,46);(3,11)(4,13)(8,14)(9,16)(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32)(34,39)(36,41)(40,46)(44,47);(2,7)(3,11)(9,16)(10,12)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,48)(46,47);(1,3)(2,8)(4,12)(5,9)(6,11)(7,14)(10,13)(15,16)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,47)(46,48);(1,4,6,13)(2,9,7,16)(3,12,11,10)(5,14,15,8)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32);(1,5)(2,10)(3,9)(4,14)(6,15)(7,12)(8,13)(11,16)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,37,38,45)(34,40,39,46)(35,42,43,48)(36,44,41,47)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000688);