Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420455 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15278 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,27,7,12,19,11)(3,13,28,31,17,39,16,8)(4,9,32,34,22,35,21,51)(5,23,25,38,26,10,36,53)(14,41,43,57,18,40,46,29)(15,37,55,58,44,52,24,30)(20,47,59,63,50,33,49,61)(42,62,54,64,45,60,48,56);(1,3)(2,8)(4,18)(5,24)(6,28)(7,17)(9,29)(10,37)(11,39)(12,31)(13,27)(14,22)(15,25)(16,19)(20,48)(21,43)(23,52)(26,55)(30,38)(32,46)(33,60)(34,41)(35,57)(36,44)(40,51)(42,59)(45,49)(47,62)(50,54)(53,58)(56,61)(63,64);(1,4,19,32,7,22,6,21)(2,9,27,51,12,35,11,34)(3,14,28,46,17,18,16,43)(5,20,36,59,26,50,25,49)(8,29,39,41,31,57,13,40)(10,33,53,63,23,47,38,61)(15,42,55,48,44,45,24,54)(30,56,52,62,58,64,37,60);(1,5,6,25,7,26,19,36)(2,10,11,38,12,23,27,53)(3,15,16,24,17,44,28,55)(4,20,21,49,22,50,32,59)(8,30,13,37,31,58,39,52)(9,33,34,61,35,47,51,63)(14,42,43,54,18,45,46,48)(29,56,40,60,57,64,41,62)Polycyclic presentation:
2^9|a^2=ef,c^2=ei,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327652);