Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #392 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1183 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32)(33,34,37,42,38,41,49,56)(35,39,45,52,44,51,58,62)(36,40,48,54,47,55,60,64)(43,50,57,61,46,53,59,63);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32)(33,35)(34,39)(36,46)(37,45)(38,44)(40,53)(41,51)(42,52)(43,47)(48,59)(49,58)(50,55)(54,63)(56,62)(57,60)(61,64);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,23)(19,25)(24,30)(26,31)(33,36)(34,40)(35,43)(37,48)(38,47)(39,50)(41,55)(42,54)(44,46)(45,57)(49,60)(51,53)(52,61)(56,64)(58,59)(62,63);(3,11)(6,9)(10,16)(13,15)(18,23)(22,28)(24,30)(29,32)(34,41)(36,47)(39,51)(42,56)(43,46)(48,60)(52,62)(57,59);(2,8)(6,9)(7,14)(13,15)(17,20,21,27)(18,24,23,30)(19,26,25,31)(22,29,28,32)(35,44)(39,51)(43,46)(45,58)(50,53)(52,62)(57,59)(61,63)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=df,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6317063);