Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2428 |
| 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 | |
1595 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)(3,14)(4,20,24,36)(5,25)(6,29)(7,12)(8,13)(9,17)(10,23,37,30)(11,26)(15,31,21,34)(16,44)(18,33)(19,41)(22,35,52,59)(27,38)(28,54)(32,47)(39,43,57,48)(40,45)(42,60,50,64)(46,61)(49,55,53,51)(56,63,58,62);(1,3,6,17,8,19,7,18)(2,9,12,14,13,33,29,41)(4,21,30,43,24,15,23,48)(5,16,26,45,28,47,27,46)(10,34,20,39,37,31,36,57)(11,32,25,40,38,44,54,61)(22,50,55,62,52,42,51,63)(35,58,49,60,59,56,53,64);(1,4)(2,10)(3,15)(5,22)(6,23)(7,30)(8,24)(9,31)(11,35)(12,36)(13,37)(14,39)(16,42)(17,43)(18,48)(19,21)(20,29)(25,53)(26,51)(27,55)(28,52)(32,56)(33,34)(38,59)(40,60)(41,57)(44,58)(45,62)(46,63)(47,50)(49,54)(61,64);(1,5,6,26,8,28,7,27)(2,11,12,25,13,38,29,54)(3,16,17,45,19,47,18,46)(4,22,23,51,24,52,30,55)(9,32,14,40,33,44,41,61)(10,35,36,53,37,59,20,49)(15,42,43,62,21,50,48,63)(31,56,39,60,34,58,57,64)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272227);