Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124974 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1517 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C2 | #7642 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
267 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 267 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)(3,11,13,27,15,28,14,26)(4,16,19,9)(5,8)(6,10)(7,12,22,17,23,29,21,30)(18,25)(20,31,32,24);(1,3)(2,7)(4,17)(5,13)(6,15)(8,21)(9,26)(10,23)(11,24)(12,20)(14,18)(16,27)(19,30)(22,25)(28,31)(29,32);(2,8,10,25)(7,21,23,22)(9,24,16,31)(11,27,28,26)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=cfgh,d^a=df,f^a=fi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32170);