Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359901 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2342 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13901 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
959 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)(3,14,18,9)(4,19)(5,24,27,11)(6,28)(7,13)(8,12)(10,22)(15,42,45,35)(16,32)(17,33,29,34)(20,31,44,40)(21,50,52,48)(23,38)(25,39,26,54)(30,37)(36,51,53,56)(41,47)(43,60)(46,59)(49,61)(55,58)(57,63)(62,64);(1,3,6,17,8,18,7,29)(2,9,28,34,12,14,13,33)(4,20,22,15,23,44,30,45)(5,16,25,46,27,47,26,55)(10,35,38,31,37,42,19,40)(11,32,54,59,24,41,39,58)(21,49,51,43,52,63,56,64)(36,60,50,57,53,62,48,61);(1,4)(2,10,12,37)(3,15)(5,21)(6,22)(7,30)(8,23)(9,31,14,40)(11,36,24,53)(13,19,28,38)(16,43)(17,44)(18,45)(20,29)(25,51)(26,56)(27,52)(32,57,41,61)(33,35,34,42)(39,48,54,50)(46,63)(47,64)(49,55)(58,60,59,62);(1,5,6,25,8,27,7,26)(2,11,28,54,12,24,13,39)(3,16,17,46,18,47,29,55)(4,21,22,51,23,52,30,56)(9,32,34,59,14,41,33,58)(10,36,38,50,37,53,19,48)(15,43,44,63,45,64,20,49)(31,57,42,62,40,61,35,60)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268763);