Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×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 | |
270 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 142 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)(17,18,22,26,24,27,23,32)(19,28,20,30,29,31,25,21);(2,5)(6,8)(17,19,21,27,23,20,31,32,24,29,30,18,22,25,28,26);(1,3,4,7)(2,6,5,8)(17,20,21,32,23,29,31,18,24,25,30,26,22,19,28,27)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,c^b=ci,f^b=fi,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72343564, #72343565, #72343568, … |
| maximal normal subgroup of | #72343564, #72343565, #72343568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42846);