Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2908 conjugacy classes of subgroups; only the 578 normal subgroups are shown.
This group has 2908 conjugacy classes of subgroups — too many to draw. Only its 578 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 — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,13,15,16)(25,26,28,29)(27,31,32,30);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,18)(19,22)(20,21)(23,24)(26,29)(30,31);(3,7)(6,8)(10,14)(13,16)(18,21)(22,24)(25,27)(26,30)(28,32)(29,31);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(26,29)(27,32);(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^9|a^2=fgi,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dfg,d^c=df92 conjugacy classes — view the full character table.
| Fitting subgroup of | #72266640 |
| maximal subgroup of | #72266595, #72266640, #72303615, … |
| maximal normal subgroup of | #72266595, #72266640, #72303615, … |
| maximal quotient of | #72266629 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 752487);