Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384421 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,14,6,12)(3,15,19,9,20,34,18,35)(4,21,25,10,26,38,24,39)(5,27,29,42,31,11,30,41)(16,43,22,36,46,32,45,59)(17,47,28,33,49,40,48,60)(23,52,53,37,55,50,54,56)(44,58,57,61,63,62,51,64);(1,3,8,20)(2,9,14,35)(4,22,26,45)(5,28,31,48)(6,19,7,18)(10,36,39,59)(11,40,27,47)(12,34,13,15)(16,24,46,25)(17,30,49,29)(21,32,38,43)(23,44,55,63)(33,42,60,41)(37,62,56,58)(50,61,52,64)(51,54,57,53);(1,4)(2,10,14,39)(3,16)(5,23)(6,24)(7,25)(8,26)(9,32,35,43)(11,37,27,56)(12,21,13,38)(15,36,34,59)(17,44)(18,45)(19,22)(20,46)(28,57)(29,53)(30,54)(31,55)(33,58,60,62)(40,61,47,64)(41,50,42,52)(48,51)(49,63);(1,5,6,29,8,31,7,30)(2,11,12,41,14,27,13,42)(3,17,18,28,20,49,19,48)(4,23,24,53,26,55,25,54)(9,33,15,40,35,60,34,47)(10,37,21,50,39,56,38,52)(16,44,45,57,46,63,22,51)(32,58,36,61,43,62,59,64)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291620);