Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| 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 | |
976 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)(4,16,19,26)(5,20)(6,10)(7,11)(8,14)(9,18,25,17)(13,28,30,22)(15,23)(21,24)(27,29,32,31)(33,34)(35,39,45,52)(36,46,48,40)(37,41)(38,42)(43,50)(44,51,57,62)(47,59,60,53)(49,54)(55,61)(56,58)(63,64);(1,3,5,14,7,15,6,21)(2,8,10,12,11,24,20,23)(4,13,18,31,19,30,17,29)(9,22,26,27,25,28,16,32)(33,35)(34,39,42,52)(36,43)(37,44)(38,45)(40,50,46,58)(41,51,54,62)(47,55)(48,56)(49,57)(53,61,59,64)(60,63);(1,4)(2,9)(3,13)(5,18)(6,17)(7,19)(8,22)(10,26)(11,25)(12,27)(14,31)(15,30)(16,20)(21,29)(23,32)(24,28)(33,36)(34,40)(35,43)(37,47)(38,48)(39,50)(41,53)(42,46)(44,55)(45,56)(49,60)(51,61)(52,58)(54,59)(57,63)(62,64);(33,37,38,49)(34,41,42,54)(35,44,45,57)(36,47,48,60)(39,51,52,62)(40,53,46,59)(43,55,56,63)(50,61,58,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fhi,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106715);