Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2408 |
| 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:
(1,2,7,12,8,13,6,29)(3,14,19,32,18,33,17,9)(4,10,23,36,24,38,22,37)(5,25,27,39,28,55,26,11)(15,41,44,30,43,34,20,49)(16,45,48,57,47,31,46,40)(21,51,53,60,54,59,52,35)(42,56,63,58,62,64,50,61);(1,3)(2,9,13,32)(4,20)(5,16)(6,17)(7,19)(8,18)(10,34,38,41)(11,31,39,45)(12,14,29,33)(15,23)(21,50)(22,43)(24,44)(25,40,55,57)(26,46)(27,48)(28,47)(30,36,49,37)(35,58,60,61)(42,53)(51,64,59,56)(52,62)(54,63);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,24)(9,30)(11,35)(12,36)(13,38)(14,34)(16,42)(17,20)(18,43)(19,44)(25,51)(26,52)(27,53)(28,54)(29,37)(31,56)(32,49)(33,41)(39,60)(40,58)(45,64)(46,50)(47,62)(48,63)(55,59)(57,61);(1,5,6,26,8,28,7,27)(2,11,12,25,13,39,29,55)(3,16,17,46,18,47,19,48)(4,21,22,52,24,54,23,53)(9,31,14,40,32,45,33,57)(10,35,36,51,38,60,37,59)(15,42,20,50,43,62,44,63)(30,56,34,58,49,64,41,61)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,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272236);