Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10,14,18)(11,16,12,15)(13,21,22,17)(19,23,20,24)(25,26,32,36)(27,37,42,53)(28,43,29,34)(30,35,31,47)(33,40,50,41)(38,54,45,49)(39,48,55,56)(44,52,46,51);(9,11)(10,15)(12,14)(13,20)(16,18)(17,24)(19,22)(21,23)(25,27,31,41,32,42,30,40)(26,33,35,37,36,50,47,53)(28,38,44,55,29,45,46,39)(34,48,52,49,43,56,51,54);(1,3,4,8)(2,5,6,7)(9,12,14,11)(10,16,18,15)(13,20,22,19)(17,24,21,23)(25,28,32,29)(26,34,36,43)(27,38,42,45)(30,46,31,44)(33,48,50,56)(35,52,47,51)(37,49,53,54)(39,41,55,40);(11,12)(15,16)(19,20)(23,24)(25,29,32,28)(26,34,36,43)(27,39,42,55)(30,46,31,44)(33,49,50,54)(35,51,47,52)(37,48,53,56)(38,41,45,40);(9,13,14,22)(10,17,18,21)(11,19,12,20)(15,23,16,24)(25,30,32,31)(26,35,36,47)(27,40,42,41)(28,44,29,46)(33,37,50,53)(34,51,43,52)(38,55,45,39)(48,54,56,49)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=g,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7095226);