Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144372 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C23×Q8 | C23 |
| prime radical | #23910 | C2 |
| LCS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
847 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 847 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(10,14,16,18)(15,24,17,23);(1,3)(2,5)(4,7)(6,8)(9,10)(11,18)(12,15)(13,16)(14,20)(17,21)(19,24)(22,23);(5,8)(10,15)(12,21)(14,23)(16,17)(18,24)(19,22)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,f^a=fghi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dg,f^c=fi,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72231752, #72238412, #72238413, … |
| maximal normal subgroup of | #72231752, #72238412, #72238413, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51602);