Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93073 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1296 |
| commutator | C4×C8 | C42 |
| Frattini, cosocle | C2×C82 | C22 |
| prime radical | C23 | #333 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
159 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 159 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,8,7,11,15,10)(3,6,12,16,9,4,13,14)(17,18,20,25,22,27,21,26)(19,28,29,32,31,23,30,24);(1,3,6,8,5,13,16,2,7,9,4,10,15,12,14,11)(18,23)(19,29)(20,21)(24,26)(25,32)(27,28)(30,31)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=h,e^2=fg,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72216576, #72216593, #72225155, … |
| maximal normal subgroup of | #72216576, #72216593, #72225155, … |
| maximal quotient of | #72216557 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272);