Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94233 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7236 |
| commutator | C2×M16 | C42 |
| Frattini, cosocle | #2931 | C22 |
| prime radical | #532 | D4 |
| LCS | G ▷ C2×M16 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #2931 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
203 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 203 conjugacy classes of subgroups — too many to draw. Only its 48 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:
(1,2,5,11)(3,15,17,52)(4,10,22,12)(6,30,27,43)(7,13,29,47)(8,33,35,46)(9,36,37,56)(14,28,42,32)(16,50,54,45)(18,58,21,39)(19,53,55,38)(20,41,57,63)(23,51,31,49)(24,44,26,48)(25,60,34,62)(40,61,64,59);(1,3,6,18,8,34,26,61,7,20,31,54,32,19,22,36)(2,9,12,38,14,50,49,63,13,40,48,60,33,39,30,52)(4,21,5,25,27,59,35,57,24,16,29,55,23,56,28,17)(10,41,42,64,51,62,47,58,44,15,46,37,43,53,11,45)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,e^2=fgi,g^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=ghi,h^a=hi,c^b=ci,d^b=dg,f^b=fi,g^b=gi,h^b=hi,d^c=dh,f^c=fi,g^c=gi,f^d=fi,g^d=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1430);