Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#106 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
280 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 512 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,11,13,12,14,15,16)(18,24,27,26)(19,29,28,30)(20,25)(21,32)(22,31);(2,5)(6,8)(17,18,21,26,22,28,32,30,23,27,20,24,31,19,25,29);(1,3,4,7)(2,6,5,8)(17,19,20,30,22,18,25,24,23,28,21,29,31,27,32,26)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=d,d^2=g,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,f^b=fi,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72343574, #72343575, #72343581, … |
| maximal normal subgroup of | #72343574, #72343575, #72343581, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42844);