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 | |
4216 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)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32)(33,34,38,37)(35,41,43,39)(36,40,45,44)(42,47,48,46);(4,11)(8,14)(10,15)(12,16)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(33,35,38,43)(34,39,37,41)(36,42,45,48)(40,46,44,47);(2,7)(3,9)(8,14)(10,15)(18,20)(19,26)(25,28)(27,32)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,35)(34,41)(36,42)(37,39)(38,43)(40,47)(44,46)(45,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,37,38,34)(35,41,43,39)(36,44,45,40)(42,47,48,46)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9887521);