Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #389324 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14539 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
912 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)(3,9,17,34)(4,19)(5,25,29,40)(6,14)(7,13)(8,12)(10,23)(11,27,39,28)(15,35,43,59)(16,44)(18,41,30,33)(20,31,50,47)(21,36,51,62)(22,37)(24,38)(26,54)(32,45)(42,63)(46,57)(48,53,61,52)(49,58)(55,56)(60,64);(1,3,6,18,8,17,7,30)(2,9,13,33,12,34,14,41)(4,20,23,43,22,50,24,15)(5,26,27,46,29,16,28,45)(10,35,19,47,38,59,37,31)(11,32,40,44,39,57,25,54)(21,49,52,42,51,64,53,55)(36,60,61,63,62,58,48,56);(1,4,6,23,8,22,7,24)(2,10,13,19,12,38,14,37)(3,15,30,50,17,43,18,20)(5,21,28,53,29,51,27,52)(9,31,41,59,34,47,33,35)(11,36,25,48,39,62,40,61)(16,42,45,64,26,55,46,49)(32,56,44,60,57,63,54,58);(1,5)(2,11)(3,16)(4,21)(6,28)(7,27)(8,29)(9,32)(10,36)(12,39)(13,25)(14,40)(15,42)(17,26)(18,46)(19,48)(20,49)(22,51)(23,53)(24,52)(30,45)(31,56)(33,54)(34,57)(35,58)(37,61)(38,62)(41,44)(43,55)(47,63)(50,64)(59,60)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^d=fi,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296520);