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 | #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)(4,21,25,10,26,36,24,37)(5,27,29,39,30,38,28,11)(16,31,44,41,45,50,22,34)(17,46,48,60,49,32,47,42)(23,35,54,56,55,59,53,51)(43,61,62,64,63,58,52,57);(1,3)(2,9,14,33)(4,22)(5,17)(6,18)(7,19)(8,20)(10,34,37,41)(11,32,39,46)(12,15,13,40)(16,25)(21,50,36,31)(23,52)(24,45)(26,44)(27,42,38,60)(28,47)(29,48)(30,49)(35,58,59,61)(43,54)(51,64,56,57)(53,63)(55,62);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,31)(11,35)(12,36)(13,21)(14,37)(15,41)(17,43)(18,22)(19,44)(20,45)(27,56)(28,53)(29,54)(30,55)(32,57)(33,50)(34,40)(38,51)(39,59)(42,61)(46,64)(47,52)(48,62)(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)(10,35,36,56,37,59,21,51)(16,43,22,52,45,63,44,62)(31,57,41,61,50,64,34,58)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,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272262);