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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18)(19,23)(20,29)(21,25)(22,24)(26,31)(27,32)(28,30)(33,34,37,41,38,42,49,56)(35,39,45,52,44,51,58,62)(36,40,48,55,47,54,60,64)(43,50,57,61,46,53,59,63);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,16)(10,11)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,35)(34,39)(36,46)(37,45)(38,44)(40,53)(41,52)(42,51)(43,47)(48,59)(49,58)(50,54)(55,63)(56,62)(57,60)(61,64);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(18,24)(19,26)(25,29)(27,28)(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);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(17,20,22,21)(18,25,24,29)(19,27,26,28)(23,30,31,32)(36,47)(40,54)(43,46)(48,60)(50,53)(55,64)(57,59)(61,63);(2,10)(7,11)(8,9)(15,16)(17,21,22,20)(18,25,24,29)(19,28,26,27)(23,30,31,32)(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,d^2=fg,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,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, 6317949);