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 | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1579 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,30)(3,14,19,33,18,41,17,9)(4,20,23,36,25,10,24,37)(5,26,28,38,29,57,27,11)(15,39,43,31,44,34,21,50)(16,45,48,61,47,32,46,40)(22,52,53,49,55,56,54,35)(42,62,63,60,64,58,51,59);(1,3)(2,9,13,33)(4,21,25,43)(5,16)(6,17)(7,19)(8,18)(10,34)(11,32,38,45)(12,14,30,41)(15,24,44,23)(20,39)(22,51,55,63)(26,40,57,61)(27,46)(28,48)(29,47)(31,36)(35,59)(37,50)(42,54,64,53)(49,60)(52,62)(56,58);(1,4)(2,10)(3,15)(5,22)(6,23)(7,24)(8,25)(9,31)(11,35)(12,36)(13,20)(14,39)(16,42)(17,43)(18,44)(19,21)(26,56)(27,53)(28,54)(29,55)(30,37)(32,58)(33,50)(34,41)(38,49)(40,60)(45,62)(46,63)(47,64)(48,51)(52,57)(59,61);(1,5,6,27,8,29,7,28)(2,11,12,26,13,38,30,57)(3,16,17,46,18,47,19,48)(4,22,23,53,25,55,24,54)(9,32,14,40,33,45,41,61)(10,35,36,56,20,49,37,52)(15,42,43,63,44,64,21,51)(31,58,39,60,50,62,34,59)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=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, 272266);