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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(2,5)(6,8)(9,10)(11,16)(12,14)(13,20)(15,19)(17,21)(18,22)(23,24)(25,26,31,32)(27,29,36,33)(28,34,39,46)(30,44,45,35)(37,41,40,47)(38,48,50,43)(42,55,52,49)(51,56,53,54);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(26,32)(27,36)(28,39)(35,44)(38,50)(41,47)(42,52)(54,56);(1,3,4,7)(2,6,5,8)(9,12,14,10)(11,16,19,15)(13,20,21,17)(18,23,24,22)(25,27,31,36)(26,33,32,29)(28,40,39,37)(30,38,45,50)(34,41,46,47)(35,48,44,43)(42,53,52,51)(49,54,55,56);(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,28)(26,34)(27,37)(29,41)(30,42)(31,39)(32,46)(33,47)(35,49)(36,40)(38,51)(43,54)(44,55)(45,52)(48,56)(50,53);(9,13,14,21)(10,17,12,20)(11,18,19,24)(15,22,16,23)(25,29,31,33)(26,27,32,36)(28,41,39,47)(30,43,45,48)(34,37,46,40)(35,38,44,50)(42,54,52,56)(49,51,55,53);(25,30,31,45)(26,35,32,44)(27,38,36,50)(28,42,39,52)(29,43,33,48)(34,49,46,55)(37,51,40,53)(41,54,47,56)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7622726);