Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198798 |
| 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) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,5,10)(3,13,17,24,16,23,15,8)(4,18,19,26,21,9,20,25)(14,27,28,22,30,32,29,31)(33,34)(35,45,49,56)(36,50,53,58)(37,43)(38,44)(39,42)(40,47,55,48)(41,52,57,51)(46,60)(54,63)(59,61)(62,64);(1,3,7,16)(2,8)(4,14,21,30)(5,15,6,17)(9,22)(10,23)(11,13)(12,24)(18,31)(19,28,20,29)(25,32)(26,27)(33,35,37,48,39,49,38,47)(34,40,43,56,42,55,44,45)(36,46,52,61,53,63,51,62)(41,54,50,64,57,60,58,59);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,22)(10,25)(11,26)(12,18)(13,27)(15,28)(16,30)(17,29)(23,32)(24,31)(33,36)(34,41)(35,46)(37,52)(38,51)(39,53)(40,54)(42,57)(43,50)(44,58)(45,59)(47,62)(48,61)(49,63)(55,60)(56,64);(1,5,7,6)(2,10,12,11)(3,15,16,17)(4,19,21,20)(8,23,24,13)(9,25,18,26)(14,28,30,29)(22,32,31,27)(34,42)(40,55)(41,57)(43,44)(45,56)(50,58)(54,60)(59,64)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=h,e^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106069);