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:
(1,2,6,12,8,13,7,32)(3,14,18,47,20,37,17,46)(4,21,24,43,26,42,31,10)(5,27,28,44,30,60,29,11)(9,33,36,19,38,53,35,16)(15,48,22,45,52,64,51,63)(23,40,49,55,25,41,59,56)(34,54,61,58,62,50,39,57);(1,3,5,19,6,18,28,53,8,20,30,16,7,17,29,33)(2,9,11,37,32,35,60,47,13,38,44,14,12,36,27,46)(4,22,25,57,31,15,49,50,26,51,23,58,24,52,59,54)(10,39,41,48,21,34,56,45,43,61,40,64,42,62,55,63);(1,4,8,26)(2,10,13,43)(3,15)(5,23,30,25)(6,31,7,24)(9,34)(11,40,44,41)(12,42,32,21)(14,45)(16,50)(17,22)(18,51)(19,54)(20,52)(27,56,60,55)(28,59,29,49)(33,58)(35,61)(36,39)(37,63)(38,62)(46,64)(47,48)(53,57)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,f^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, #72307242, #72307268, … |
| maximal normal subgroup of | #72307202, #72307242, #72307268, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54637);