Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151130 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12090 |
| commutator, Frattini, cosocle | #379 | C23 |
| prime radical | #23895 | C2 |
| LCS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C24 ◁ #379 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
688 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 38 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 — 3 generators on 32 points:
(1,2,6,10)(3,11,14,27)(4,8,16,23)(5,9,18,25)(7,19,21,13)(12,26,29,32)(15,24,17,22)(20,30,31,28);(1,3)(2,7)(4,12)(5,13)(6,14)(8,20)(9,11)(10,21)(15,30)(16,29)(17,28)(18,19)(22,32)(23,31)(24,26)(25,27);(1,4)(2,8)(3,12)(5,17)(6,16)(7,20)(9,24)(10,23)(11,26)(13,28)(14,29)(15,18)(19,30)(21,31)(22,25)(27,32)Polycyclic presentation:
2^9|a^2=i,d^2=f,b^a=bd,c^a=ce,d^a=df,g^a=gh,d^b=df,e^b=egh,g^b=gi,h^b=hi,d^c=dg,f^c=fi,e^d=ehi,g^d=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72283443, #72283453, #72283518, … |
| maximal normal subgroup of | #72283443, #72283453, #72283518, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58327);