Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94417 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7302 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C4×M32 | C22 |
| prime radical | C23 | #360 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C4×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
130 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 130 conjugacy classes of subgroups — too many to draw. Only its 50 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)(3,17,19,43,23,52,22,61)(4,24,27,48)(5,31,32,46)(6,35)(7,15)(8,14)(9,16)(10,18,38,25,42,53,41,54)(11,28,45,34)(12,29,47,30)(13,33)(20,40,57,63,58,49,55,64)(21,59,56,39,36,62,60,51)(26,50)(37,44);(1,3,5,20,33,56,29,18,7,19,28,55,6,36,27,54,9,23,32,58,37,60,30,53,8,22,34,57,26,21,4,25)(2,10,12,39,35,63,31,52,14,38,48,59,13,49,11,43,16,42,47,51,50,64,46,17,15,41,24,62,44,40,45,61)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=gh,e^2=fgi,f^2=h,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=di,e^d=ei50 conjugacy classes — view the full character table.
| maximal subgroup of | #72286048, #72286095, #72286111, … |
| maximal normal subgroup of | #72286048, #72286095, #72286111, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1614);