Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476849 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17098 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
853 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,8,12,7,13)(3,15,18,33,20,9,19,34)(4,21,24,10,25,43,30,37)(5,11,27,42,28,40,29,41)(16,46,22,44,48,35,51,31)(17,45,26,57,49,32,50,38)(23,52,54,39,56,62,55,36)(47,63,58,59,64,61,53,60);(1,3,8,20)(2,9,12,15)(4,22,25,51)(5,26,28,50)(6,19,7,18)(10,35,37,46)(11,38,40,57)(13,34,14,33)(16,30,48,24)(17,29,49,27)(21,31,43,44)(23,47,56,64)(32,41,45,42)(36,60,39,59)(52,61,62,63)(53,55,58,54);(1,4)(2,10,12,37)(3,16)(5,23)(6,24)(7,30)(8,25)(9,31,15,44)(11,39,40,36)(13,21,14,43)(17,47)(18,22)(19,51)(20,48)(26,58)(27,54)(28,56)(29,55)(32,60,45,59)(33,35,34,46)(38,63,57,61)(41,52,42,62)(49,64)(50,53);(1,5)(2,11,12,40)(3,17)(4,23)(6,29)(7,27)(8,28)(9,32,15,45)(10,36,37,39)(13,42,14,41)(16,47)(18,50)(19,26)(20,49)(21,52,43,62)(22,53)(24,55)(25,56)(30,54)(31,59,44,60)(33,38,34,57)(35,61,46,63)(48,64)(51,58)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,g^c=gi,f^d=fi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384046);