Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7440 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1353 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #423 | C23 |
| socle | C2 | C2×#1353 |
| prime radical | C24 | #104 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#423 ◁ G | |
| LECS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
291 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 291 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12,13,25,16,29,15,32)(4,9)(5,10,19,26)(6,21)(7,11)(8,14,22,30,23,31,28,18)(17,24,20,27);(1,3)(2,8,11,23)(4,13)(5,18,19,30)(6,15)(7,16)(9,22,21,28)(10,25)(12,27)(14,20,31,17)(24,29)(26,32);(1,4,7,6)(2,9,11,21)(3,13,16,15)(5,17,19,20)(8,22,23,28)(10,24,26,27)(12,25,29,32)(14,30,31,18)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=ghi,e^2=hi,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,d^b=dg,e^b=efgh,h^b=hi,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72339762, #72339769, #72339771 |
| maximal normal subgroup of | #72339762, #72339769, #72339771 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57505);