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 | |
4214 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)(19,25)(23,26)(27,31)(30,32)(34,39)(36,41)(37,45)(42,48);(2,7)(5,13)(8,9)(12,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(34,39)(36,41)(37,45)(42,48);(3,10)(8,9)(11,14)(12,16)(18,22)(23,26)(24,28)(30,32)(33,34,38,39)(35,41,43,36)(37,40,45,46)(42,47,48,44);(1,3)(2,8)(4,14)(5,12)(6,10)(7,9)(11,15)(13,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,38,43)(34,36,39,41)(37,42,45,48)(40,44,46,47);(1,4)(2,5)(3,11)(6,15)(7,13)(8,16)(9,12)(10,14)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,47)(46,48);(1,5)(2,4)(3,12)(6,13)(7,15)(8,14)(9,11)(10,16)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)(33,37,38,45)(34,40,39,46)(35,42,43,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,f^2=i,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, 8782734);