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 | |
4015 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,19,17)(12,20,21,15)(18,23,24,22)(25,26)(27,29)(28,34)(30,44)(31,32)(33,35)(36,45)(37,41)(38,48)(39,46)(40,47)(42,55)(43,50)(49,52)(51,56)(53,54);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(26,32)(27,35)(28,39)(36,44)(38,50)(41,47)(42,52)(54,56);(1,2,5,4)(3,6,8,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,27)(26,33)(28,40)(29,32)(30,38)(31,35)(34,41)(36,48)(37,39)(42,53)(43,44)(45,50)(46,47)(49,54)(51,52)(55,56);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,28,31,39)(26,34,32,46)(27,37,35,40)(29,41,33,47)(30,42,45,52)(36,49,44,55)(38,51,50,53)(43,54,48,56);(1,4,5,2)(3,7,8,6)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,31,33)(26,35,32,27)(28,41,39,47)(30,43,45,48)(34,40,46,37)(36,50,44,38)(42,54,52,56)(49,53,55,51);(25,30,31,45)(26,36,32,44)(27,38,35,50)(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=gi,b^2=gh,c^2=gh,d^2=h,e^2=g,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, 7622708);