Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94514 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7338 |
| commutator | C25 | C2×C8 |
| Frattini, cosocle | #1781 | C22 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C25 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G | |
| LECS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
346 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 24 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 — 2 generators on 32 points:
(1,2,5,9)(3,11,13,25)(4,15,16,30,17,8,18,23)(6,10,19,24)(7,20,21,12,22,27,31,28)(14,26,29,32);(1,3,6,14)(2,7,10,22)(4,12,17,28)(5,13,19,29)(8,11,15,26)(9,21,24,31)(16,27,18,20)(23,25,30,32)Polycyclic presentation:
2^9|a^2=d,b^2=i,d^2=f,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,d^b=de,f^b=fh,d^c=dg,f^c=fi,e^d=eh,g^d=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72283518, #72283597, #72283614, … |
| maximal normal subgroup of | #72283518, #72283597, #72283614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1711);