Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13215 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1961 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10927 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,12,8,14,26,13)(3,15,19,45,18,44,49,46)(4,10,23,38,24,41,58,40)(5,25,27,11,7,29,28,42)(9,31,34,20,33,52,35,17)(16,47,21,54,51,43,56,64)(22,57,59,37,30,60,48,39)(32,61,63,50,62,55,36,53);(1,3,7,20,26,49,27,31,8,18,5,17,6,19,28,52)(2,9,11,15,12,34,29,45,14,33,42,44,13,35,25,46)(4,21,22,55,58,16,48,50,24,56,30,61,23,51,59,53)(10,36,39,64,38,32,57,47,41,63,37,54,40,62,60,43);(1,4,6,23,8,24,26,58)(2,10,12,38,14,41,13,40)(3,16,49,56,18,51,19,21)(5,22,27,59,7,30,28,48)(9,32,35,36,33,62,34,63)(11,37,29,60,42,39,25,57)(15,43,46,54,44,47,45,64)(17,50,52,61,20,53,31,55)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,c^b=ch,e^b=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307219, #72307245, #72307258, … |
| maximal normal subgroup of | #72307219, #72307245, #72307258, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54622);