Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7535047 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3549 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | #59945 |
| prime radical | #24288 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1152 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)(3,14)(4,20)(5,25)(6,10)(7,13)(8,17)(9,24)(11,27)(12,28)(15,29)(16,44)(18,32)(19,31)(21,39)(22,36)(23,52)(26,37)(30,45)(33,43)(34,50)(35,51)(38,54)(40,46)(41,56)(42,60)(47,57)(48,53)(49,63)(55,62)(58,61)(59,64);(1,3,6,17,7,19,28,18)(2,8,12,14,13,32,10,31)(4,21,24,15,22,43,51,41)(5,16,27,46,26,45,54,47)(9,33,20,29,35,39,36,56)(11,30,25,40,38,44,37,57)(23,49,53,42,50,62,64,61)(34,58,48,55,52,60,59,63);(1,4,7,22)(2,9)(3,15)(5,23,26,50)(6,24,28,51)(8,29,32,56)(10,36)(11,34)(12,20)(13,35)(14,39,31,33)(16,42)(17,43)(18,21)(19,41)(25,48)(27,53,54,64)(30,55,44,63)(37,59)(38,52)(40,60,57,58)(45,61)(46,62)(47,49);(2,10,13,12)(4,22)(5,26)(8,14,32,31)(9,20,35,36)(11,25,38,37)(15,41)(16,45)(21,43)(24,51)(27,54)(29,33,56,39)(30,57,44,40)(34,59,52,48)(46,47)(55,60,63,58);(1,5,6,27,7,26,28,54)(2,11,12,25,13,38,10,37)(3,16,17,46,19,45,18,47)(4,23,24,53,22,50,51,64)(8,30,14,40,32,44,31,57)(9,34,20,48,35,52,36,59)(15,42,43,62,41,61,21,49)(29,55,39,60,56,63,33,58)Polycyclic presentation:
2^9|b^2=f,c^2=fgh,d^2=h,e^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529602);