Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6370843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#117 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2283 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #2873 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1685 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)(3,13,16,35)(4,18,21,9)(5,10)(6,11)(7,12)(8,17,27,42)(14,36)(15,34,41,55)(19,44,46,28)(20,45,48,29)(22,30)(23,31)(24,32)(25,39)(26,40,51,60)(33,38)(37,56)(43,59)(47,62,63,52)(49,53)(50,57)(54,58)(61,64);(1,3)(2,8,12,27)(4,14)(5,15)(6,17)(7,16)(9,25,18,43)(10,26,30,51)(11,13,32,35)(19,37)(20,39)(21,38)(22,41)(23,40)(24,42)(28,50,44,61)(29,33,45,36)(31,34,53,55)(46,58)(47,57)(48,59)(49,60)(52,54,62,56)(63,64);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,25)(10,28)(11,29)(12,18)(13,33)(15,37)(16,38)(17,39)(22,46)(23,47)(24,48)(26,50)(27,43)(30,44)(31,52)(32,45)(34,54)(35,36)(40,57)(41,58)(42,59)(49,63)(51,61)(53,62)(55,56)(60,64);(1,5,6,23,7,22,24,49)(2,10,11,31,12,30,32,53)(3,15,17,40,16,41,42,60)(4,19,20,47,21,46,48,63)(8,26,13,34,27,51,35,55)(9,28,29,52,18,44,45,62)(14,37,39,57,38,58,59,64)(25,50,33,54,43,61,36,56);(3,16)(5,22)(8,27)(10,30)(13,35)(14,38)(17,42)(19,46)(23,49)(25,43)(28,44)(31,53)(33,36)(39,59)(47,63)(52,62)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=f,f^2=i,b^a=bf,c^a=cg,g^a=gi,e^b=ei,h^b=hi,h^c=hi,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278300);