Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D8 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C2×#335 |
| prime radical | C23 | #335 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
168 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 94 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 — 3 generators on 32 points:
(2,6)(3,7)(4,10)(8,12)(9,16)(11,15)(13,14)(17,18,19,23,20,24,25,30,21,22,26,28,27,29,31,32);(1,2,7,14,4,9,11,12,5,8,15,16,10,13,3,6);(2,8)(6,12)(9,13)(14,16)(18,22)(23,28)(24,29)(30,32)Polycyclic presentation:
2^9|a^2=e,b^2=df,d^2=fh,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,c^b=ch104 conjugacy classes — view the full character table.
| maximal subgroup of | #72225391, #72226932, #72226933, … |
| maximal normal subgroup of | #72225391, #72226932, #72226933, … |
| maximal quotient of | #72225375 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30131);