Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200737 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,9,13,24)(3,15,17,49)(4,20)(5,23)(6,29)(10,36,40,63)(11,42,43,32)(12,44,21,34)(14,35,33,59)(16,51,25,61)(18,30,19,50)(22,57)(27,45)(28,53)(31,37,52,47)(38,46,60,39)(41,62,56,64)(48,55,58,54);(1,2,5,24,26,33,53,35)(3,10,25,60,17,56,16,52)(4,21,22,34,29,43,57,32)(6,11,27,42,20,12,45,44)(7,14,23,59,8,13,28,9)(15,47,58,63,49,39,48,62)(18,40,55,38,19,41,54,31)(30,37,61,36,50,46,51,64);(1,3)(2,10)(4,15)(5,25)(6,30)(7,18)(8,19)(9,31)(11,37)(12,46)(13,41)(14,40)(16,53)(17,26)(20,50)(21,47)(22,58)(23,55)(24,60)(27,61)(28,54)(29,49)(32,62)(33,56)(34,63)(35,52)(36,42)(38,59)(39,43)(44,64)(45,51)(48,57);(1,4,7,6)(2,11,14,21)(3,15,18,30)(5,22,23,27)(8,20,26,29)(9,32,35,44)(10,37,40,47)(12,13,43,33)(16,48,54,51)(17,49,19,50)(24,42,59,34)(25,58,55,61)(28,45,53,57)(31,62,52,64)(36,38,63,60)(39,56,46,41)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,f^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fh,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107935);