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 | |
5260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,5)(3,7)(10,13)(11,15)(17,18,23,27)(19,24,28,37)(20,32,33,25)(21,22,26,36)(29,43,41,38)(30,31,39,42)(34,46,40,35)(44,48,47,45);(2,5)(6,8)(9,10,12,13)(11,14,15,16)(19,28)(20,33)(24,37)(25,32)(30,39)(31,42)(34,40)(35,46);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,28)(18,24,27,37)(20,29,33,41)(21,30,26,39)(22,31,36,42)(25,38,32,43)(34,44,40,47)(35,45,46,48);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,38)(26,40)(27,32)(28,41)(30,44)(31,45)(36,46)(37,43)(39,47)(42,48);(17,21)(18,22)(19,30)(20,34)(23,26)(24,31)(25,35)(27,36)(28,39)(29,44)(32,46)(33,40)(37,42)(38,45)(41,47)(43,48);(17,22)(18,26)(19,31)(20,35)(21,27)(23,36)(24,39)(25,40)(28,42)(29,45)(30,37)(32,34)(33,46)(38,47)(41,48)(43,44)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=hi,d^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7566223);