Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14569 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,31)(3,14,17,9,19,34,18,35)(4,20,23,38,24,10,25,39)(5,26,29,42,30,41,28,11)(15,36,46,60,45,43,21,32)(16,47,27,40,48,33,49,61)(22,52,54,37,55,57,56,50)(44,59,58,62,53,63,51,64);(1,3)(2,9)(4,21,24,46)(5,27)(6,18)(7,17)(8,19)(10,36,20,43)(11,40)(12,34)(13,35)(14,31)(15,25,45,23)(16,28)(22,53,55,44)(26,33)(29,48)(30,49)(32,39,60,38)(37,63,50,59)(41,47)(42,61)(51,56,58,54)(52,62,57,64);(1,4)(2,10,13,20)(3,15,19,45)(5,22)(6,25)(7,23)(8,24)(9,32)(11,37,42,50)(12,39,31,38)(14,43)(16,44,48,53)(17,46,18,21)(26,57,41,52)(27,58,49,51)(28,56)(29,54)(30,55)(33,59)(34,36)(35,60)(40,64)(47,63)(61,62);(1,5,6,28,8,30,7,29)(2,11,12,26,13,42,31,41)(3,16,17,27,19,48,18,49)(4,22,25,56,24,55,23,54)(9,33,14,40,35,47,34,61)(10,37,39,57,20,50,38,52)(15,44,46,58,45,53,21,51)(32,59,43,64,60,63,36,62)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=efh,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297535);