Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2424 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14195 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1074 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)(3,9,16,33)(4,18)(5,11,23,39)(6,26)(7,13)(8,12)(10,21)(14,34,41,57)(15,43)(17,32,27,31)(19,46,42,29)(20,47,49,59)(22,37)(24,51,25,38)(28,36)(30,45)(35,53,60,50)(40,61)(44,55)(48,54)(52,56)(58,63)(62,64);(1,3,6,17,8,16,7,27)(2,9,13,31,12,33,26,32)(4,19,21,14,22,42,28,41)(5,15,24,44,23,45,25,52)(10,34,18,46,36,57,37,29)(11,30,38,55,39,43,51,56)(20,48,50,40,49,62,53,63)(35,58,47,64,60,61,59,54);(1,4,6,21,8,22,7,28)(2,10,13,18,12,36,26,37)(3,14,17,42,16,41,27,19)(5,20,24,50,23,49,25,53)(9,29,31,34,33,46,32,57)(11,35,38,47,39,60,51,59)(15,40,44,62,45,63,52,48)(30,54,55,58,43,64,56,61);(1,5)(2,11,12,39)(3,15)(4,20)(6,24)(7,25)(8,23)(9,30,33,43)(10,35,36,60)(13,38,26,51)(14,40)(16,45)(17,44)(18,47,37,59)(19,48)(21,50)(22,49)(27,52)(28,53)(29,54,46,64)(31,55,32,56)(34,58,57,61)(41,63)(42,62)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=eh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284616);