Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1313 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)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,32,37)(27,33,41,51)(28,35,29,43)(30,36,31,46)(34,44,53,42)(38,48,39,54)(40,49,45,50)(47,55,52,56);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,32,41)(26,33,37,51)(28,39,29,38)(30,45,31,40)(34,52,53,47)(35,54,43,48)(36,50,46,49)(42,55,44,56);(11,17)(14,21)(18,23)(22,24)(25,28,32,29)(26,34,37,53)(27,38,41,39)(30,42,31,44)(33,47,51,52)(35,36,43,46)(40,55,45,56)(48,50,54,49);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,29,32,28)(26,35,37,43)(27,39,41,38)(30,42,31,44)(33,48,51,54)(34,36,53,46)(40,55,45,56)(47,50,52,49);(1,3,4,7)(2,6,5,8)(25,30,32,31)(26,36,37,46)(27,40,41,45)(28,42,29,44)(33,49,51,50)(34,43,53,35)(38,55,39,56)(47,48,52,54)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,e^b=ei,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343522);