Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #391213 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14621 |
| 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 | |
1062 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)(3,14,18,42,19,9,17,32)(4,20,22,10,23,36,28,35)(5,11,25,41,26,39,29,40)(15,44,46,33,47,51,49,30)(16,43,24,57,48,31,50,37)(21,52,56,34,55,62,54,38)(45,61,58,60,64,63,53,59);(1,3,8,19)(2,9,12,14)(4,15,23,47)(5,24,26,50)(6,18,7,17)(10,33,35,30)(11,37,39,57)(13,42,27,32)(16,29,48,25)(20,51,36,44)(21,53,55,58)(22,49,28,46)(31,40,43,41)(34,61,38,63)(45,56,64,54)(52,60,62,59);(1,4)(2,10)(3,15)(5,21)(6,28)(7,22)(8,23)(9,30)(11,38)(12,35)(13,36)(14,33)(16,45)(17,49)(18,46)(19,47)(20,27)(24,58)(25,56)(26,55)(29,54)(31,60)(32,44)(34,39)(37,63)(40,62)(41,52)(42,51)(43,59)(48,64)(50,53)(57,61);(1,5)(2,11)(3,16)(4,21)(6,29)(7,25)(8,26)(9,31)(10,34)(12,39)(13,41)(14,43)(15,45)(17,50)(18,24)(19,48)(20,52)(22,56)(23,55)(27,40)(28,54)(30,59)(32,37)(33,60)(35,38)(36,62)(42,57)(44,61)(46,58)(47,64)(49,53)(51,63)Polycyclic presentation:
2^9|a^2=h,b^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,f^c=fi,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298412);