Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1367 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)(3,7)(4,6)(5,8)(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,26,32,37)(27,33,42,50)(28,43,29,35)(30,36,31,46)(34,45,51,44)(38,52,39,48)(40,49,41,55)(47,54,56,53);(1,3,4,8)(2,5,6,7)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(25,27,32,42)(26,33,37,50)(28,38,29,39)(30,40,31,41)(34,47,51,56)(35,48,43,52)(36,49,46,55)(44,53,45,54);(3,8)(5,7)(9,10)(11,15)(12,21)(13,16)(14,17)(18,20)(19,24)(22,23)(25,28,32,29)(26,34,37,51)(27,38,42,39)(30,44,31,45)(33,47,50,56)(35,46,43,36)(40,53,41,54)(48,55,52,49);(2,6)(5,7)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,29,32,28)(26,35,37,43)(27,39,42,38)(30,44,31,45)(33,48,50,52)(34,46,51,36)(40,53,41,54)(47,55,56,49);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,30,32,31)(26,36,37,46)(27,40,42,41)(28,44,29,45)(33,49,50,55)(34,35,51,43)(38,53,39,54)(47,48,56,52)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6893798);