Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #385905 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14417 |
| 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 | |
1358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,6,13)(3,15,19,33,20,9,18,34)(4,21,25,10,26,38,24,37)(5,27,28,11,30,41,29,40)(16,43,22,42,46,35,45,31)(17,47,48,39,50,57,49,32)(23,53,54,51,55,36,56,58)(44,59,52,61,63,62,64,60);(1,3,8,20)(2,9,12,15)(4,22,26,45)(5,17,30,50)(6,19,7,18)(10,35,37,43)(11,39,40,32)(13,34,14,33)(16,24,46,25)(21,31,38,42)(23,52,55,64)(27,57,41,47)(28,49,29,48)(36,61,53,60)(44,56,63,54)(51,62,58,59);(1,4,6,25,8,26,7,24)(2,10,13,38,12,37,14,21)(3,16,18,22,20,46,19,45)(5,23,29,54,30,55,28,56)(9,31,33,43,15,42,34,35)(11,36,27,58,40,53,41,51)(17,44,49,52,50,63,48,64)(32,59,57,61,39,62,47,60);(1,5)(2,11)(3,17)(4,23)(6,29)(7,28)(8,30)(9,32)(10,36)(12,40)(13,27)(14,41)(15,39)(16,44)(18,49)(19,48)(20,50)(21,51)(22,52)(24,56)(25,54)(26,55)(31,59)(33,57)(34,47)(35,60)(37,53)(38,58)(42,62)(43,61)(45,64)(46,63)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=eg,f^2=i,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,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293104);