Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187204 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62829 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,5,10)(3,13,17,23,16,8,15,24)(4,18,19,26,20,9,21,25)(14,27,28,32,29,22,30,31)(33,34)(35,43)(36,47,49,40)(37,50)(38,41)(39,46)(42,52)(44,58,59,57)(45,56)(48,55,64,62)(51,54)(53,60,61,63);(1,3)(2,8)(4,14)(5,17)(6,15)(7,16)(9,22)(10,24)(11,13)(12,23)(18,27)(19,30)(20,29)(21,28)(25,32)(26,31)(33,35)(34,39)(36,44)(37,51)(38,45)(40,53)(41,54)(42,56)(43,50)(46,52)(47,61)(48,63)(49,59)(55,58)(57,62)(60,64);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,25)(11,18)(12,26)(13,27)(15,28)(16,29)(17,30)(23,31)(24,32)(33,36)(34,40)(35,44)(37,48)(38,49)(39,53)(41,47)(42,55)(43,57)(45,59)(46,60)(50,62)(51,63)(52,64)(54,61)(56,58);(1,5,7,6)(2,10,11,12)(3,15,16,17)(4,19,20,21)(8,23,13,24)(9,25,18,26)(14,28,29,30)(22,31,27,32)(34,41)(35,45)(40,47)(42,50)(44,59)(46,51)(55,62)(60,63)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94579);