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 | |
4050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(29,42)(33,46)(37,51)(40,53)(44,48)(45,52)(49,56)(54,55);(1,2)(3,5)(4,6)(7,8)(25,26,30,34)(27,35,38,31)(28,32,41,39)(29,33,42,46)(36,47,50,43)(37,48,51,44)(40,45,53,52)(49,55,56,54);(1,3,4,7)(2,5,6,8)(10,14)(15,17)(16,21)(22,23)(25,26)(27,31)(28,39)(29,33)(30,34)(32,41)(35,38)(36,47)(37,44)(40,52)(42,46)(43,50)(45,53)(48,51)(49,55)(54,56);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27,30,38)(26,31,34,35)(28,36,41,50)(29,37,42,51)(32,43,39,47)(33,44,46,48)(40,49,53,56)(45,54,52,55);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28)(26,32)(27,36)(29,40)(30,41)(31,43)(33,45)(34,39)(35,47)(37,49)(38,50)(42,53)(44,54)(46,52)(48,55)(51,56);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29)(26,33)(27,37)(28,40)(30,42)(31,44)(32,45)(34,46)(35,48)(36,49)(38,51)(39,52)(41,53)(43,54)(47,55)(50,56)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=i,e^2=h,f^2=h,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, 7621677);