Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18346 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
989 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)(3,12,14,24)(4,16)(5,10)(6,20)(7,11)(8,15,23,19)(9,18)(13,28,31,22)(17,26)(21,25)(27,29,32,30)(33,34)(35,39)(36,46,48,40)(37,49)(38,42)(41,50)(43,51,56,59)(44,57)(45,53)(47,54,61,60)(52,58)(55,63,64,62);(1,3,5,19,7,14,6,15)(2,8,10,24,11,23,20,12)(4,13,18,30,17,31,21,29)(9,22,26,32,25,28,16,27)(33,35,38,45)(34,39,42,53)(36,43,48,56)(37,44,50,58)(40,51,46,59)(41,52,49,57)(47,55,61,64)(54,62,60,63);(1,4)(2,9)(3,13)(5,18)(6,21)(7,17)(8,22)(10,26)(11,25)(12,27)(14,31)(15,29)(16,20)(19,30)(23,28)(24,32)(33,36)(34,40)(35,43)(37,47)(38,48)(39,51)(41,54)(42,46)(44,55)(45,56)(49,60)(50,61)(52,62)(53,59)(57,63)(58,64);(3,14)(4,17)(8,23)(9,25)(12,24)(15,19)(16,26)(18,21)(33,37,38,50)(34,41,42,49)(35,44,45,58)(36,47,48,61)(39,52,53,57)(40,54,46,60)(43,55,56,64)(51,62,59,63)Polycyclic presentation:
2^9|a^2=g,b^2=egi,d^2=i,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=di,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102302);