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 | #14576 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
787 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)(3,14,18,34)(4,19)(5,24,28,42)(6,12)(7,31)(8,13)(9,17,35,29)(10,23)(11,26,41,27)(15,36,47,43)(16,33)(20,32,46,60)(21,52,55,57)(22,39)(25,40)(30,38)(37,54,50,56)(44,49)(45,64)(48,61)(51,62)(53,59)(58,63);(1,3)(2,9)(4,20,23,46)(5,25)(6,29)(7,17)(8,18)(10,36,19,43)(11,40)(12,34)(13,35)(14,31)(15,22,47,30)(16,26)(21,53,55,45)(24,33)(27,48)(28,49)(32,39,60,38)(37,59,50,64)(41,44)(42,61)(51,56,58,54)(52,62,57,63);(1,4,7,22,8,23,6,30)(2,10,12,39,13,19,31,38)(3,15,29,46,18,47,17,20)(5,21,26,56,28,55,27,54)(9,32,14,43,35,60,34,36)(11,37,42,52,41,50,24,57)(16,45,25,58,48,53,49,51)(33,59,44,63,61,64,40,62);(1,5)(2,11)(3,16)(4,21)(6,27)(7,26)(8,28)(9,33)(10,37)(12,42)(13,41)(14,44)(15,45)(17,49)(18,48)(19,50)(20,51)(22,56)(23,55)(24,31)(25,29)(30,54)(32,59)(34,40)(35,61)(36,62)(38,57)(39,52)(43,63)(46,58)(47,53)(60,64)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^c=ei,f^c=fi,e^d=ei,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297782);