Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93139 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1304 |
| commutator | C25 | C42 |
| Frattini, prime radical, cosocle | C22×#121 | C22 |
| LCS | G ▷ C25 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G | |
| LECS | G ▷ C22×#121 ▷ C24 ▷ C22 ▷ C1 | |
507 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 507 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 64 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,8,5,7,12,15)(3,6,9,13,10,14,11,16)(17,18,20,23)(19,25,26,31)(21,24,29,32)(22,27,30,28);(1,3)(2,6,7,14)(4,11,12,9)(5,10)(8,13)(15,16)(17,19)(18,22,24,31)(20,28,29,26)(21,27)(23,25)(30,32)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=h,b^a=bc,c^a=cf,e^a=eg,f^a=fh,g^a=gi,c^b=cg,d^b=df,f^b=fi,g^b=ghi,d^c=dh,e^c=ehi,e^d=ei32 conjugacy classes — view the full character table.
| maximal subgroup of | #72235243, #72235245, #72235247, … |
| maximal normal subgroup of | #72235243, #72235245, #72235247, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336);