Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,6,4,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,30,34)(27,31,38,44)(28,32,40,46)(29,33,36,42)(35,41,48,52)(37,43,50,54)(39,45,49,53)(47,51,55,56);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(27,35)(28,39)(29,36)(31,41)(32,45)(33,42)(37,50)(38,48)(40,49)(43,54)(44,52)(46,53);(1,4,5,3)(2,6,7,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,36,48,30,38,29,35)(26,31,42,52,34,44,33,41)(28,37,49,55,40,50,39,47)(32,43,53,56,46,54,45,51);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(28,40)(32,46)(37,50)(39,49)(43,54)(45,53)(47,55)(51,56);(11,21)(15,18)(19,20)(23,24)(25,28,29,39,30,40,36,49)(26,32,33,45,34,46,42,53)(27,37,35,47,38,50,48,55)(31,43,41,51,44,54,52,56)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=fghi,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6727753);