Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24808 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2416 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13872 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1543 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)(3,15)(4,20,24,10)(5,25)(6,14)(7,12)(8,13)(9,18)(11,26)(16,41,45,34)(17,46)(19,39)(21,50,44,31)(22,52,54,55)(23,36,30,40)(27,37)(28,38)(29,33)(32,48)(35,53,49,57)(42,47)(43,62,64,58)(51,59,63,61)(56,60);(1,3,6,18,8,19,7,29)(2,9,12,15,13,33,14,39)(4,21,23,16,24,44,30,45)(5,17,26,47,28,48,27,56)(10,34,36,31,20,41,40,50)(11,32,25,42,37,46,38,60)(22,51,53,43,54,63,57,64)(35,59,55,58,49,61,52,62);(1,4)(2,10)(3,16)(5,22)(6,23)(7,30)(8,24)(9,31)(11,35)(12,36)(13,20)(14,40)(15,41)(17,43)(18,44)(19,45)(21,29)(25,55)(26,53)(27,57)(28,54)(32,58)(33,50)(34,39)(37,49)(38,52)(42,61)(46,62)(47,63)(48,64)(51,56)(59,60);(1,5,6,26,8,28,7,27)(2,11,12,25,13,37,14,38)(3,17,18,47,19,48,29,56)(4,22,23,53,24,54,30,57)(9,32,15,42,33,46,39,60)(10,35,36,55,20,49,40,52)(16,43,44,63,45,64,21,51)(31,58,41,61,50,62,34,59)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267427);