Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383745 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14363 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,27)(3,14,19,41,17,9,18,32)(4,20,24,35,23,10,28,36)(5,11,26,40,25,38,29,39)(15,42,21,33,44,30,47,49)(16,31,45,57,46,37,48,56)(22,34,53,60,54,50,52,61)(43,62,64,59,63,55,51,58);(1,3)(2,9)(4,21)(5,16)(6,19)(7,18)(8,17)(10,33)(11,37)(12,14)(13,41)(15,24)(20,49)(22,51)(23,47)(25,46)(26,48)(27,32)(28,44)(29,45)(30,35)(31,38)(34,59)(36,42)(39,56)(40,57)(43,52)(50,58)(53,63)(54,64)(55,61)(60,62);(1,4)(2,10,12,20)(3,15)(5,22)(6,28)(7,24)(8,23)(9,30,14,42)(11,34,38,50)(13,36,27,35)(16,43)(17,44)(18,47)(19,21)(25,54)(26,53)(29,52)(31,55,37,62)(32,49,41,33)(39,61,40,60)(45,64)(46,63)(48,51)(56,59,57,58);(1,5,8,25)(2,11)(3,16,17,46)(4,22,23,54)(6,26,7,29)(9,31)(10,34)(12,38)(13,39)(14,37)(15,43,44,63)(18,45,19,48)(20,50)(21,51,47,64)(24,52,28,53)(27,40)(30,55)(32,56)(33,58)(35,60)(36,61)(41,57)(42,62)(49,59)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=ei,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,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290944);