Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324500 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1321 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,32)(27,37,40,51)(28,34,41,47)(29,36,31,35)(33,44,46,39)(38,53,42,56)(43,52,45,50)(48,54,49,55);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(26,32)(28,41)(33,46)(35,36)(37,51)(38,42)(43,45)(54,55);(1,3)(2,5)(4,8)(6,7)(25,27,29,44,30,40,31,39)(26,33,35,51,32,46,36,37)(28,42,45,55,41,38,43,54)(34,49,52,53,47,48,50,56);(2,6)(3,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28,30,41)(26,34,32,47)(27,38,40,42)(29,45,31,43)(33,48,46,49)(35,52,36,50)(37,53,51,56)(39,55,44,54);(2,6)(5,7)(10,16)(14,17)(15,24)(22,23)(25,29,30,31)(26,35,32,36)(27,39,40,44)(28,43,41,45)(33,37,46,51)(34,50,47,52)(38,54,42,55)(48,56,49,53)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dhi,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7236109);