Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1936 |
| 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 | |
685 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)(3,14)(4,20,24,42)(5,26)(6,12)(7,32)(8,13)(9,16)(10,23,40,31)(11,28)(15,47,51,64)(17,45)(18,37)(19,35)(21,44,50,63)(22,39,25,41)(27,43)(29,60)(30,46)(33,58,61,57)(34,52)(36,53)(38,54,62,49)(48,55,59,56);(1,3,5,18,6,30,27,52,8,19,29,16,7,17,28,53)(2,9,11,35,32,34,60,46,13,37,43,14,12,36,26,45)(4,21,25,57,31,15,48,49,24,50,22,58,23,51,59,54)(10,38,41,47,20,33,56,44,40,62,39,64,42,61,55,63);(1,4,6,31,8,24,7,23)(2,10,12,42,13,40,32,20)(3,15,17,21,19,51,30,50)(5,22,27,59,29,25,28,48)(9,33,34,62,37,61,36,38)(11,39,26,56,43,41,60,55)(14,44,45,64,35,63,46,47)(16,49,52,57,18,54,53,58)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ei,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307202, #72307216, #72307268, … |
| maximal normal subgroup of | #72307202, #72307216, #72307268, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54634);