Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,30,32)(27,36,37,51)(28,41,43,34)(29,35,31,45)(33,39,46,40)(38,53,52,50)(42,49,44,48)(47,54,56,55);(1,2,4,5)(3,7,8,6)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(26,32)(27,37)(34,41)(35,45)(38,52)(39,40)(48,49)(54,55);(2,5)(6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,29,39,30,37,31,40)(26,33,45,51,32,46,35,36)(28,38,44,55,43,52,42,54)(34,47,48,53,41,56,49,50);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,28,30,43)(26,34,32,41)(27,38,37,52)(29,44,31,42)(33,47,46,56)(35,49,45,48)(36,50,51,53)(39,55,40,54);(1,3,4,8)(2,6,5,7)(11,21)(15,18)(19,20)(23,24)(25,29,30,31)(26,35,32,45)(27,39,37,40)(28,42,43,44)(33,36,46,51)(34,48,41,49)(38,54,52,55)(47,53,56,50)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=fg,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6711942);