Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14358 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,14)(3,15,19,34,20,42,18,9)(4,21,24,37,26,10,25,38)(5,27,30,11,31,40,29,41)(16,35,22,52,46,43,50,32)(17,47,48,44,49,33,28,39)(23,54,55,36,56,58,57,51)(45,60,59,63,64,62,53,61);(1,3,7,19,8,20,6,18)(2,9,14,42,13,34,12,15)(4,22,25,16,26,50,24,46)(5,28,30,17,31,48,29,49)(10,35,38,52,21,43,37,32)(11,39,27,33,41,44,40,47)(23,45,57,53,56,64,55,59)(36,62,58,61,51,60,54,63);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,32)(11,36)(12,37)(13,21)(14,38)(15,43)(17,45)(18,22)(19,50)(20,46)(27,58)(28,59)(29,55)(30,57)(31,56)(33,60)(34,52)(35,42)(39,61)(40,54)(41,51)(44,63)(47,62)(48,53)(49,64);(1,5)(2,11)(3,17)(4,23)(6,30)(7,29)(8,31)(9,33)(10,36)(12,27)(13,41)(14,40)(15,44)(16,45)(18,48)(19,28)(20,49)(21,51)(22,53)(24,57)(25,55)(26,56)(32,60)(34,47)(35,61)(37,58)(38,54)(39,42)(43,63)(46,64)(50,59)(52,62)Polycyclic presentation:
2^9|a^2=h,b^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,e^d=ei,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290383);