Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622467 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C23×#99 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C2×Q8 | C25 |
| prime radical | #62859 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×D4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
4073 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,11,7,12,27,38)(3,13,17,40,15,35,25,52)(4,9,20,34,18,32,46,57)(5,21,23,37,24,10,26,36)(8,28,31,22,30,44,56,16)(14,39,43,61,41,58,50,63)(19,47,49,60,45,33,51,59)(29,53,55,48,54,62,64,42);(1,3)(2,8)(4,14)(5,22)(6,25)(7,15)(9,29)(10,35)(11,31)(12,30)(13,21)(16,24)(17,27)(18,41)(19,48)(20,50)(23,28)(26,44)(32,54)(33,58)(34,55)(36,52)(37,40)(38,56)(39,47)(42,45)(43,46)(49,53)(51,62)(57,64)(59,63)(60,61);(1,4)(2,9)(3,14)(5,19)(6,20)(7,18)(8,29)(10,33)(11,34)(12,32)(13,39)(15,41)(16,42)(17,43)(21,47)(22,48)(23,49)(24,45)(25,50)(26,51)(27,46)(28,53)(30,54)(31,55)(35,58)(36,59)(37,60)(38,57)(40,61)(44,62)(52,63)(56,64);(4,18)(9,32)(14,41)(19,45)(20,46)(29,54)(33,47)(34,57)(39,58)(42,48)(43,50)(49,51)(53,62)(55,64)(59,60)(61,63);(3,15)(8,30)(13,35)(14,41)(16,22)(17,25)(28,44)(29,54)(31,56)(39,58)(40,52)(42,48)(43,50)(53,62)(55,64)(61,63)Polycyclic presentation:
2^9|a^2=g,f^2=i,g^2=i,b^a=bf,f^a=fh,h^a=hi,e^b=ei,f^b=fi,g^b=ghi,d^c=di,g^f=gi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529679);