Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465309 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16518 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
741 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)(4,18,22,9)(5,23,26,10)(6,27,29,11)(8,30,33,16)(14,40,19,51)(15,45,48,41)(20,50,53,35)(21,36,28,39)(24,37,58,55)(25,38,59,56)(31,44,34,49)(32,47,54,60)(43,64,52,62)(46,63,57,61);(1,3,6,16,7,17,29,30)(2,8,11,13,12,33,27,42)(4,19,28,44,22,14,21,49)(5,15,25,47,26,48,59,60)(9,34,39,51,18,31,36,40)(10,32,38,41,23,54,56,45)(20,52,58,46,53,43,24,57)(35,63,55,62,50,61,37,64);(1,4,6,28,7,22,29,21)(2,9,11,39,12,18,27,36)(3,14,30,44,17,19,16,49)(5,24,25,20,26,58,59,53)(8,31,42,51,33,34,13,40)(10,37,38,35,23,55,56,50)(15,46,60,52,48,57,47,43)(32,62,45,63,54,64,41,61);(1,5,6,25,7,26,29,59)(2,10,11,38,12,23,27,56)(3,15,16,47,17,48,30,60)(4,20,21,24,22,53,28,58)(8,32,13,41,33,54,42,45)(9,35,36,37,18,50,39,55)(14,43,44,46,19,52,49,57)(31,61,51,62,34,63,40,64)Polycyclic presentation:
2^9|a^2=i,b^2=eg,c^2=ef,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,f^c=fi,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, 372506);