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 | #53449 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4055 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)(3,5)(4,7)(6,8)(10,15)(13,21)(16,22)(20,24)(25,26)(27,33)(28,34)(29,35)(30,36)(31,32)(37,38)(39,48)(40,46)(41,49)(42,50)(43,51)(44,52)(45,47)(53,54)(55,56);(1,2,6,8)(3,5,7,4)(12,19)(16,22)(18,23)(20,24)(26,32)(29,34)(33,46)(36,47)(38,49)(43,50)(48,52)(54,56);(1,3,6,7)(2,4,8,5)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,27)(26,33)(28,41)(29,38)(30,44)(31,40)(32,46)(34,49)(35,37)(36,52)(39,45)(42,53)(43,56)(47,48)(50,54)(51,55);(1,4,6,5)(2,3,8,7)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,28)(26,34)(27,37)(29,32)(30,42)(31,35)(33,38)(36,50)(39,53)(40,41)(43,47)(44,55)(45,51)(46,49)(48,54)(52,56);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,31,34)(26,35,32,28)(27,38,40,49)(30,43,45,50)(33,37,46,41)(36,51,47,42)(39,54,44,56)(48,53,52,55);(1,5,6,4)(2,7,8,3)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,30,31,45)(26,36,32,47)(27,39,40,44)(28,42,35,51)(29,43,34,50)(33,48,46,52)(37,53,41,55)(38,54,49,56)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839281);