Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632048 |
| 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 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1399 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)(11,17)(12,20)(13,16)(14,19)(15,22)(18,23)(21,24)(25,26,29,33)(27,35,37,31)(28,39,41,50)(30,44,43,34)(32,45,49,42)(36,52,40,56)(38,47,54,51)(46,53,48,55);(1,3)(2,5)(4,8)(6,7)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,27,29,37)(26,31,33,35)(28,40,41,36)(30,38,43,54)(32,48,49,46)(34,47,44,51)(39,52,50,56)(42,55,45,53);(3,8)(5,7)(25,28,29,41)(26,32,33,49)(27,36,37,40)(30,45,43,42)(31,46,35,48)(34,50,44,39)(38,55,54,53)(47,52,51,56);(2,6)(5,7)(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(26,33)(31,35)(32,49)(34,44)(39,50)(46,48)(47,51)(52,56);(10,16)(14,17)(15,20)(23,24)(27,37)(31,35)(36,40)(38,54)(46,48)(47,51)(52,56)(53,55)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,c^2=i,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6555019);