Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60470 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#121 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C22×#389 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12,8,13,6,28)(3,14,17,9,19,31,18,32)(4,20,23,34,24,48,22,10)(5,11,27,36,25,35,26,52)(15,29,40,46,42,54,41,37)(16,38,44,30,43,56,45,55)(21,47,51,57,49,63,50,33)(39,53,60,62,59,64,61,58);(1,3,7,17,8,19,6,18)(2,9,12,31,13,32,28,14)(4,15,23,40,24,42,22,41)(5,16,27,44,25,43,26,45)(10,29,20,46,34,54,48,37)(11,30,36,56,35,55,52,38)(21,39,51,60,49,59,50,61)(33,53,47,62,57,64,63,58);(1,4,6,22,8,24,7,23)(2,10,12,20,13,34,28,48)(3,15,18,41,19,42,17,40)(5,21,26,50,25,49,27,51)(9,29,31,46,32,54,14,37)(11,33,36,47,35,57,52,63)(16,39,45,61,43,59,44,60)(30,53,56,62,55,64,38,58);(1,5)(2,11)(3,16)(4,21)(6,26)(7,27)(8,25)(9,30)(10,33)(12,36)(13,35)(14,38)(15,39)(17,44)(18,45)(19,43)(20,47)(22,50)(23,51)(24,49)(28,52)(29,53)(31,56)(32,55)(34,57)(37,58)(40,60)(41,61)(42,59)(46,62)(48,63)(54,64);(5,25)(11,35)(16,43)(21,49)(26,27)(30,55)(33,57)(36,52)(38,56)(39,59)(44,45)(47,63)(50,51)(53,64)(58,62)(60,61)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,f^b=fi,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279111);