Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #362481 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2293 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13921 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,13,6,12)(3,15,18,33,20,34,19,9)(4,21,24,40,25,10,30,37)(5,26,27,38,29,39,28,11)(16,42,44,58,45,35,22,31)(17,46,48,41,49,32,47,59)(23,52,54,50,55,56,53,36)(43,57,51,60,62,64,63,61);(1,3)(2,9,13,33)(4,22,25,44)(5,17)(6,18)(7,19)(8,20)(10,35)(11,32,38,46)(12,15,14,34)(16,24,45,30)(21,42)(23,51,55,63)(26,41,39,59)(27,47)(28,48)(29,49)(31,40)(36,60)(37,58)(43,53,62,54)(50,61)(52,57)(56,64);(1,4,6,24,8,25,7,30)(2,10,12,37,13,21,14,40)(3,16,18,22,20,45,19,44)(5,23,27,53,29,55,28,54)(9,31,15,35,33,58,34,42)(11,36,26,56,38,50,39,52)(17,43,47,51,49,62,48,63)(32,57,41,60,46,64,59,61);(1,5,6,27,8,29,7,28)(2,11,14,39,13,38,12,26)(3,17,18,47,20,49,19,48)(4,23,24,53,25,55,30,54)(9,32,34,59,33,46,15,41)(10,36,40,52,21,50,37,56)(16,43,22,51,45,62,44,63)(31,57,42,61,58,64,35,60)Polycyclic presentation:
2^9|a^2=h,b^2=efg,c^2=e,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269884);