Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #361285 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2290 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13901 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
969 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)(3,15,19,39)(4,10,23,36)(5,24,26,37)(6,12)(7,14)(8,13)(9,18,32,28)(11,25,38,27)(16,41)(17,46)(20,30)(21,50)(22,35,29,40)(31,48)(33,45)(34,54)(42,53)(43,62,64,55)(44,56)(47,59)(49,61,63,57)(51,58)(52,60);(1,3,8,28,7,19,6,18)(2,9,12,15,14,32,13,39)(4,20,29,45,23,44,22,16)(5,17,27,53,26,48,25,47)(10,33,35,30,36,41,40,56)(11,31,24,42,38,46,37,59)(21,49,54,64,52,63,51,43)(34,57,50,55,58,61,60,62);(1,4,6,22,7,23,8,29)(2,10,12,35,14,36,13,40)(3,16,18,44,19,45,28,20)(5,21,25,51,26,52,27,54)(9,30,15,41,32,56,39,33)(11,34,24,50,38,58,37,60)(17,43,47,63,48,64,53,49)(31,55,42,61,46,62,59,57);(1,5,8,27,7,26,6,25)(2,11,12,24,14,38,13,37)(3,17,28,53,19,48,18,47)(4,21,29,54,23,52,22,51)(9,31,15,42,32,46,39,59)(10,34,35,50,36,58,40,60)(16,43,20,49,45,64,44,63)(30,55,41,61,56,62,33,57)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268757);