Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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)(27,36,37,53)(28,41,42,34)(29,35,31,46)(33,39,47,40)(38,54,43,52)(44,51,45,50)(48,55,49,56);(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)(28,42)(35,46)(39,40)(44,45)(48,49)(52,54);(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)(26,33,46,53,32,47,35,36)(28,43,45,55,42,38,44,56)(34,49,50,52,41,48,51,54);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,28,30,42)(26,34,32,41)(27,38,37,43)(29,45,31,44)(33,48,47,49)(35,51,46,50)(36,52,53,54)(39,56,40,55);(1,3,4,8)(2,6,5,7)(11,21)(15,18)(19,20)(23,24)(25,29,30,31)(26,35,32,46)(27,39,37,40)(28,44,42,45)(33,36,47,53)(34,50,41,51)(38,55,43,56)(48,54,49,52)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,d^b=di,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6712057);