Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C22⋊C4 |
| commutator | C2×C4 | C4×C16 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1374 |
| prime radical | C23 | C4⋊C16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
150 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 72 points:
(2,6)(3,4)(7,8)(9,10,12,17,13,19,26,37,14,20,27,38,29,41,50,61,15,18,28,36,30,39,51,59,31,40,52,60,53,62,68,71)(11,21,23,42,24,43,47,63,25,44,48,64,49,65,67,72,22,16,32,33,45,34,54,56,46,35,55,57,66,58,69,70);(1,2,4,7,5,8,3,6)(9,11)(10,16,18,21)(12,23)(13,24)(14,25)(15,22)(17,33,36,42)(19,34,39,43)(20,35,40,44)(26,47)(27,48)(28,32)(29,49)(30,45)(31,46)(37,56,59,63)(38,57,60,64)(41,58,62,65)(50,67)(51,54)(52,55)(53,66)(61,70,71,72)(68,69)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=f,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efi,c^b=ci152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1213);