Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1361 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)(25,26,31,43,30,35,29,34)(27,32,44,54,37,46,38,47)(28,39,40,33,42,48,41,49)(36,50,52,45,51,56,53,55);(2,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,27)(26,32)(28,36)(29,38)(30,37)(31,44)(33,45)(34,47)(35,46)(39,50)(40,52)(41,53)(42,51)(43,54)(48,56)(49,55);(1,3,4,7)(2,6,5,8)(10,14)(15,21)(16,22)(23,24)(25,28,30,42)(26,33,35,49)(27,36,37,51)(29,40,31,41)(32,45,46,55)(34,48,43,39)(38,52,44,53)(47,56,54,50);(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(27,37)(32,46)(36,51)(38,44)(45,55)(47,54)(50,56)(52,53);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,29,30,31)(26,34,35,43)(27,38,37,44)(28,40,42,41)(32,47,46,54)(33,48,49,39)(36,52,51,53)(45,56,55,50)Polycyclic presentation:
2^9|a^2=gi,c^2=fi,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6457614);