Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1246 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)(25,26,31,33,29,35,30,36)(27,37,45,49,39,32,40,47)(28,41,46,52,43,34,44,50)(38,48,54,55,42,51,53,56);(1,3,7,4)(2,5,8,6)(9,10)(11,18)(12,15)(13,16)(14,17)(19,23)(20,22)(21,24)(25,27,29,39)(26,32,35,37)(28,42,43,38)(30,40,31,45)(33,47,36,49)(34,51,41,48)(44,54,46,53)(50,56,52,55);(1,4,7,3)(2,5,8,6)(9,11,13,17)(10,14,16,18)(12,20,21,19)(15,23,24,22)(26,33)(28,43)(30,31)(32,47)(34,52)(35,36)(37,49)(38,42)(40,45)(41,50)(48,56)(51,55);(1,5,7,6)(2,3,8,4)(9,12,13,21)(10,15,16,24)(11,19,17,20)(14,22,18,23)(25,28)(26,34)(27,38)(29,43)(30,44)(31,46)(32,48)(33,50)(35,41)(36,52)(37,51)(39,42)(40,53)(45,54)(47,55)(49,56);(1,6,7,5)(2,3,8,4)(10,16)(14,18)(15,24)(22,23)(27,39)(28,43)(32,37)(34,41)(40,45)(44,46)(47,49)(50,52)Polycyclic presentation:
2^9|a^2=ghi,b^2=f,c^2=fhi,d^2=fhi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6870524);