Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499867 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #62817 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4922 conjugacy classes of subgroups; only the 598 normal subgroups are shown.
This group has 4922 conjugacy classes of subgroups — too many to draw. Only its 598 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 — 5 generators on 24 points:
(9,10,13,16)(11,14,18,21)(12,15,19,22)(17,23,24,20);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,17)(13,18)(15,20)(16,21)(19,24)(22,23);(1,3)(2,5)(4,8)(6,7)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,24)(21,23);(3,8)(5,7);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72234625, #72235079, #72235716, … |
| maximal normal subgroup of | #72234625, #72235079, #72235716, … |
| maximal quotient of | #72234626, #72251070, #72284555, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407064);