Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465212 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16516 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1053 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,13,17,42)(4,9,21,36)(5,22,25,10)(6,26,28,11)(8,29,32,16)(14,43,18,40)(15,46,49,41)(19,34,52,63)(20,39,27,35)(23,54,57,37)(24,38,58,55)(30,50,33,45)(31,48,53,59)(44,61,51,64)(47,60,56,62);(1,3,6,16,7,17,28,29)(2,8,11,13,12,32,26,42)(4,18,27,45,21,14,20,50)(5,15,24,48,25,49,58,59)(9,33,39,43,36,30,35,40)(10,31,38,41,22,53,55,46)(19,51,57,47,52,44,23,56)(34,62,54,61,63,60,37,64);(1,4)(2,9)(3,14,17,18)(5,23)(6,27)(7,21)(8,30,32,33)(10,37)(11,39)(12,36)(13,40,42,43)(15,47,49,56)(16,50,29,45)(19,24)(20,28)(22,54)(25,57)(26,35)(31,61,53,64)(34,38)(41,60,46,62)(44,59,51,48)(52,58)(55,63);(1,5,6,24,7,25,28,58)(2,10,11,38,12,22,26,55)(3,15,16,48,17,49,29,59)(4,19,20,23,21,52,27,57)(8,31,13,41,32,53,42,46)(9,34,35,37,36,63,39,54)(14,44,45,47,18,51,50,56)(30,60,43,61,33,62,40,64)Polycyclic presentation:
2^9|a^2=i,b^2=eg,c^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372409);