Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6407635 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2380 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60645 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1968 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)(3,13,17,8)(4,18,22,46)(5,23)(6,12)(7,11)(9,21,34,51)(10,25)(14,31)(15,42,43,52)(16,38,26,30)(19,28)(20,33,49,60)(24,37)(27,32)(29,44,35,55)(36,54)(39,48)(40,61)(41,57)(45,63,53,50)(47,58)(56,62)(59,64);(1,3)(2,8)(4,19)(5,15)(6,26)(7,17)(9,31)(10,35)(11,13)(12,38)(14,21)(16,27)(18,39)(20,47)(22,48)(23,52)(24,43)(25,55)(28,46)(29,36)(30,32)(33,59)(34,57)(37,42)(40,50)(41,51)(44,54)(45,56)(49,64)(53,61)(58,60)(62,63);(1,4,6,21,7,22,27,51)(2,9,32,46,11,34,12,18)(3,14,16,19,17,41,26,48)(5,20,25,50,24,49,54,63)(8,28,38,31,13,39,30,57)(10,33,23,53,36,60,37,45)(15,40,44,47,43,62,55,64)(29,56,52,58,35,61,42,59);(1,5)(2,10,11,36)(3,15)(4,20)(6,25)(7,24)(8,29,13,35)(9,33,34,60)(12,37,32,23)(14,40)(16,44)(17,43)(18,45,46,53)(19,47)(21,50)(22,49)(26,55)(27,54)(28,56,39,61)(30,42,38,52)(31,58,57,59)(41,62)(48,64)(51,63);(5,24)(10,36)(15,43)(20,49)(23,37)(25,54)(29,35)(33,60)(40,62)(42,52)(44,55)(45,53)(47,64)(50,63)(56,61)(58,59)Polycyclic presentation:
2^9|a^2=h,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,c^b=cfg,d^b=df,g^b=gi,h^b=hi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314831);