Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1287 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,30,35,29,34,42,32)(27,36,39,44,38,31,43,46)(28,33,41,49,40,48,53,47)(37,50,52,55,51,45,54,56);(1,2,4,5)(3,7,8,6)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,29,38)(26,31,34,36)(28,37,40,51)(30,39,42,43)(32,44,35,46)(33,45,48,50)(41,52,53,54)(47,55,49,56);(1,2)(3,6)(4,5)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(30,42)(31,44)(33,47)(34,35)(36,46)(39,43)(41,53)(45,55)(48,49)(50,56)(52,54);(2,5)(6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,29,40)(26,33,34,48)(27,37,38,51)(30,41,42,53)(31,45,36,50)(32,47,35,49)(39,52,43,54)(44,55,46,56);(1,3,4,8)(2,6,5,7)(10,18)(15,19)(16,17)(23,24)(26,34)(28,40)(31,36)(32,35)(37,51)(41,53)(44,46)(52,54)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,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, 7392336);