Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360368 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2344 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13918 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
751 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)(3,15,20,9)(4,21,25,40)(5,26,29,11)(6,13)(7,14)(8,12)(10,24,37,30)(16,43)(17,32)(18,34,19,33)(22,31)(23,52)(27,39,28,38)(35,45)(36,53)(41,49)(42,48)(44,60,63,62)(46,58)(47,59)(50,54)(51,61,64,57)(55,56);(1,3)(2,9)(4,22,25,49)(5,17)(6,18)(7,19)(8,20)(10,35,37,43)(11,32)(12,15)(13,33)(14,34)(16,24,45,30)(21,41,40,31)(23,51,54,64)(26,42)(27,46)(28,47)(29,48)(36,60,55,62)(38,58)(39,59)(44,53,63,56)(50,61,52,57);(1,4)(2,10,12,37)(3,16)(5,23)(6,24)(7,30)(8,25)(9,31,15,41)(11,36,26,55)(13,21,14,40)(17,44)(18,22)(19,49)(20,45)(27,53)(28,56)(29,54)(32,57,42,61)(33,35,34,43)(38,50,39,52)(46,51)(47,64)(48,63)(58,60,59,62);(1,5,6,27,8,29,7,28)(2,11,13,38,12,26,14,39)(3,17,18,46,20,48,19,47)(4,23,24,53,25,54,30,56)(9,32,33,58,15,42,34,59)(10,36,21,50,37,55,40,52)(16,44,22,51,45,63,49,64)(31,57,35,60,41,61,43,62)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=e,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269763);