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 | |
4119 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18,22,19)(20,25,28,23)(21,26,29,24)(27,30,32,31)(33,34,40,45)(35,43,37,41)(36,42,38,47)(39,44,48,46);(2,7)(5,8)(9,12)(13,15)(17,18,22,19)(20,23,28,25)(21,26,29,24)(27,31,32,30)(33,35,40,37)(34,41,45,43)(36,46,38,44)(39,42,48,47);(2,7)(3,10)(9,12)(11,16)(18,19)(23,25)(24,26)(30,31)(33,36,40,38)(34,42,45,47)(35,46,37,44)(39,41,48,43);(1,3,6,10)(2,8,7,5)(4,11,14,16)(9,15,12,13)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,37,40,35)(34,43,45,41)(36,46,38,44)(39,42,48,47);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,38,40,36)(34,42,45,47)(35,46,37,44)(39,43,48,41);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,16,12,11)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,39,40,48)(34,44,45,46)(35,47,37,42)(36,41,38,43)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662364);