Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #403354 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14873 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2456 conjugacy classes of subgroups; only the 280 normal subgroups are shown.
This group has 2456 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)(5,9)(6,10)(7,14)(8,17)(12,21)(13,20)(16,28)(18,23)(19,26)(22,29)(24,27)(25,30)(31,32);(1,3)(2,7,10,11)(4,12)(5,13)(6,14)(8,21,15,19)(9,20,23,24)(16,25)(17,26)(18,27)(22,31,28,30)(29,32);(1,4)(2,8,10,15)(3,12)(5,16)(6,17)(7,19,11,21)(9,22,23,28)(13,25)(14,26)(18,29)(20,30,24,31)(27,32);(1,5)(2,9)(3,13)(4,16)(6,18)(7,20)(8,22)(10,23)(11,24)(12,25)(14,27)(15,28)(17,29)(19,30)(21,31)(26,32)Polycyclic presentation:
2^9|b^2=g,c^2=g,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,f^c=fi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237894, #72237901, #72237914, … |
| maximal normal subgroup of | #72237894, #72237901, #72237914, … |
| maximal quotient of | #72265184, #72265943 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310550);