Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #28182 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13970 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1599 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,19,33,20,40,18,9)(4,21,24,10,26,36,25,37)(5,27,29,39,30,38,28,11)(16,31,44,34,45,57,22,41)(17,46,48,60,49,32,47,42)(23,52,53,50,55,35,54,59)(43,61,62,56,63,58,51,64);(1,3)(2,9,14,33)(4,22,26,44)(5,17)(6,18)(7,19)(8,20)(10,34)(11,32,39,46)(12,15,13,40)(16,25,45,24)(21,31)(23,51,55,62)(27,42,38,60)(28,47)(29,48)(30,49)(35,58)(36,57)(37,41)(43,54,63,53)(50,56)(52,61)(59,64);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,31)(11,35)(12,21)(13,36)(14,37)(15,41)(17,43)(18,44)(19,22)(20,45)(27,50)(28,53)(29,54)(30,55)(32,56)(33,57)(34,40)(38,59)(39,52)(42,61)(46,64)(47,62)(48,51)(49,63)(58,60);(1,5,6,28,8,30,7,29)(2,11,12,27,14,39,13,38)(3,17,18,47,20,49,19,48)(4,23,24,53,26,55,25,54)(9,32,15,42,33,46,40,60)(10,35,21,50,37,52,36,59)(16,43,44,62,45,63,22,51)(31,56,41,61,57,64,34,58)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272240);