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 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3737 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)(5,13)(6,7)(10,15)(11,12)(14,16)(18,23)(19,26)(25,28)(27,32)(33,34,38,42)(35,39,43,47)(36,40,46,45)(37,41,48,44);(1,2)(3,8)(4,9)(5,10)(6,7)(11,14)(12,16)(13,15)(17,18)(19,24)(20,28)(21,26)(22,23)(25,30)(27,31)(29,32)(35,43)(36,46)(39,47)(40,45);(2,7)(3,9)(10,13)(11,16)(33,35,38,43)(34,39,42,47)(36,44,46,41)(37,45,48,40);(1,3,6,9)(2,8,7,4)(5,11,15,16)(10,14,13,12)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,29,28,31);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,13,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,31,24,29)(33,36,38,46)(34,40,42,45)(35,44,43,41)(37,47,48,39);(1,5)(2,10)(3,11)(4,12)(6,15)(7,13)(8,14)(9,16)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,27,28,32)(33,37)(34,41)(35,45)(36,39)(38,48)(40,43)(42,44)(46,47)Polycyclic presentation:
2^9|a^2=hi,c^2=hi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9758261);