Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7447 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1361 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| prime radical | C23 | #361 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
148 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 68 points:
(1,2,3,4)(5,6)(7,18)(8,25)(9,30)(10,34)(11,17)(12,19)(13,27)(14,43)(15,26)(16,32)(20,38)(21,57)(22,35)(23,40)(24,50)(28,63)(29,44)(31,65)(33,41)(36,62)(37,53)(39,54)(42,67)(45,60)(46,56)(47,61)(48,52)(49,51)(55,68)(58,59)(64,66);(5,7,8,20,26,55,60,66,10,23,28,19,33,21,43,56,11,24,29,53,61,51,67,52,32,22,58,54,9,31,27,62)(6,12,14,35,30,64,63,49,16,38,45,50,47,36,25,57,17,39,13,40,41,48,59,68,34,37,42,18,15,46,44,65)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,e^2=fg,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1555);