Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7313 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1304 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C22×#131 | C22 |
| socle | C2 | #7032 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#131 ▷ C22×C4 ▷ C2 ▷ C1 | |
395 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 395 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,10,7,12,25,35,6,11,24,34,26,36,51,58)(3,13,15,37,16,39,43,59,17,38,22,48,44,60,49,63)(4,18,19,33,21,45,46,57,20,9,23,31,47,32,50,56)(8,27,28,41,30,52,54,40,29,42,53,61,55,62,64,14);(1,3)(2,8)(4,14,20,40)(5,22)(6,17)(7,16)(9,13,18,38)(10,28)(11,29)(12,30)(15,24)(19,42,23,27)(21,41,47,61)(25,49)(26,44)(31,37,33,48)(32,39,45,60)(34,53)(35,54)(36,55)(43,51)(46,62,50,52)(56,59,57,63)(58,64)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=h,h^2=i,b^a=bc,c^a=cf,e^a=egi,f^a=fi,c^b=cg,d^b=dfi,g^b=gi,d^c=di,e^c=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 677);