Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #403332 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14873 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2408 conjugacy classes of subgroups; only the 280 normal subgroups are shown.
This group has 2408 conjugacy classes of subgroups — too many to draw. Only its 280 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 — 4 generators on 32 points:
(1,2)(3,11)(4,15)(5,18)(6,10)(7,13)(8,17)(9,19)(12,24)(14,25)(16,23)(20,26)(21,28)(22,31)(27,32)(29,30);(1,3)(2,7,10,21)(4,12)(5,13)(6,14)(8,22,15,20)(9,11,18,25)(16,26)(17,27)(19,28)(23,32,29,24)(30,31);(1,4)(2,8,10,15)(3,12)(5,16)(6,17)(7,20,21,22)(9,23,18,29)(11,24,25,32)(13,26)(14,27)(19,30)(28,31);(5,19)(9,18)(11,25)(13,28)(16,30)(23,29)(24,32)(26,31)Polycyclic presentation:
2^9|b^2=g,c^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,f^c=fi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72251414, #72251419, #72251731, … |
| maximal normal subgroup of | #72251414, #72251419, #72251731, … |
| maximal quotient of | #72255848 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310528);