Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500593 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24152 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13310 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
781 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)(3,15)(4,21)(5,13)(6,29)(7,31)(8,14)(9,18)(10,23)(11,27)(12,30)(16,47)(17,45)(19,54)(20,46)(22,58)(24,42)(25,57)(26,38)(28,43)(32,39)(33,62)(34,52)(35,51)(36,53)(37,55)(40,60)(41,61)(44,48)(49,63)(50,56)(59,64);(1,3,6,18,8,20,30,53)(2,9,12,15,14,36,29,46)(4,22,25,49,26,16,24,50)(5,17,27,51,28,52,7,19)(10,37,40,44,41,33,39,64)(11,34,13,35,31,45,43,54)(21,56,42,47,38,63,57,58)(23,59,32,62,61,48,60,55);(1,4)(2,10)(3,16)(5,23)(6,24)(7,32)(8,26)(9,33)(11,38)(12,39)(13,42)(14,41)(15,44)(17,48)(18,49)(19,55)(20,22)(21,31)(25,30)(27,60)(28,61)(29,40)(34,56)(35,58)(36,37)(43,57)(45,63)(46,64)(47,54)(50,53)(51,62)(52,59);(1,5,6,27,8,28,30,7)(2,11,12,13,14,31,29,43)(3,17,18,51,20,52,53,19)(4,23,24,60,26,61,25,32)(9,34,15,35,36,45,46,54)(10,38,39,42,41,21,40,57)(16,48,49,62,22,59,50,55)(33,56,44,58,37,63,64,47)Polycyclic presentation:
2^9|b^2=h,d^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,g^b=gi,f^c=fh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407789);