Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885129 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1234 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)(3,6)(4,5)(7,8)(11,17)(14,18)(19,20)(22,23)(25,26,30,34,29,33,42,48)(27,35,38,51,37,31,44,46)(28,39,41,47,40,49,43,32)(36,52,54,45,53,56,55,50);(1,3,7,4)(2,5,8,6)(9,10)(11,18)(12,15)(13,16)(14,17)(19,23)(20,22)(21,24)(25,27,29,37)(26,31,33,35)(28,36,40,53)(30,38,42,44)(32,45,47,50)(34,46,48,51)(39,56,49,52)(41,54,43,55);(1,4,7,3)(2,5,8,6)(9,11,13,17)(10,14,16,18)(12,20,21,19)(15,23,24,22)(25,28)(26,32)(27,36)(29,40)(30,43)(31,45)(33,47)(34,49)(35,50)(37,53)(38,55)(39,48)(41,42)(44,54)(46,52)(51,56);(1,5,7,6)(2,3,8,4)(9,12,13,21)(10,15,16,24)(11,19,17,20)(14,22,18,23)(26,33)(27,37)(28,40)(34,48)(38,44)(41,43)(45,50)(52,56);(1,6,7,5)(2,3,8,4)(10,16)(14,18)(15,24)(22,23)(26,33)(27,37)(32,47)(34,48)(36,53)(38,44)(39,49)(54,55)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=fhi,d^2=fhi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6870540);