Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33691 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3109 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #21503 |
| prime radical | #33594 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
661 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,13,9,16,6,15)(3,17,22,37,23,36,20,10)(4,24,26,43,28,56,32,41)(5,12,7,31,30,14,29,42)(11,25,39,27,40,57,38,58)(18,47,49,55,51,61,54,62)(19,45,21,35,53,46,52,34)(33,50,59,63,60,64,44,48);(1,3,6,20,9,23,8,22)(2,10,13,17,16,37,15,36)(4,18,26,49,28,51,32,54)(5,19,29,52,30,53,7,21)(11,33,38,44,40,60,39,59)(12,34,31,45,14,35,42,46)(24,55,41,47,56,62,43,61)(25,48,27,50,57,63,58,64);(1,4)(2,11)(3,18)(5,27)(6,26)(7,25)(8,32)(9,28)(10,33)(12,41)(13,38)(14,43)(15,39)(16,40)(17,44)(19,50)(20,49)(21,48)(22,54)(23,51)(24,42)(29,57)(30,58)(31,56)(34,47)(35,61)(36,59)(37,60)(45,62)(46,55)(52,63)(53,64);(1,5,8,7,9,30,6,29)(2,12,13,31,16,14,15,42)(3,19,22,21,23,53,20,52)(4,25,26,27,28,57,32,58)(10,34,17,45,37,35,36,46)(11,24,39,43,40,56,38,41)(18,48,49,50,51,63,54,64)(33,55,59,61,60,62,44,47)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=g,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^c=dh,f^c=fhi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285481, #72285521, #72285937 |
| maximal normal subgroup of | #72285481, #72285521, #72285937 |
| maximal quotient of | #72287339, #72287440, #72287608, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403049);