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 | |
4283 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,8)(4,5,9,15)(12,13,14,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,32)(28,31)(29,30)(33,34,39,43)(35,37,42,40)(36,38,46,41)(44,47,48,45);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,14)(15,16)(17,19)(18,21)(20,27)(22,25)(23,30)(24,26)(28,29)(31,32)(33,35,39,42)(34,40,43,37)(36,44,46,48)(38,45,41,47);(3,11)(7,8)(12,14)(13,16)(20,28)(23,31)(27,29)(30,32)(33,36,39,46)(34,41,43,38)(35,44,42,48)(37,45,40,47);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,12)(15,16)(17,20,22,28)(18,23,26,31)(19,27,25,29)(21,30,24,32)(33,36)(34,38)(35,44)(37,47)(39,46)(40,45)(41,43)(42,48);(1,4)(2,5)(3,12)(6,9)(7,13)(8,16)(10,15)(11,14)(17,19)(18,24)(20,29)(21,26)(22,25)(23,30)(27,28)(31,32)(33,37)(34,42)(35,43)(36,45)(38,44)(39,40)(41,48)(46,47);(1,5)(2,9)(3,13)(4,10)(6,15)(7,14)(8,12)(11,16)(17,21,22,24)(18,25,26,19)(20,30,28,32)(23,29,31,27)(33,38)(34,36)(35,45)(37,48)(39,41)(40,44)(42,47)(43,46)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9080343);