Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233541 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2139 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 2139 conjugacy classes of subgroups — too many to draw. Only its 206 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)(3,7)(4,8)(5,9)(6,11)(10,17)(12,26)(13,20)(14,19)(15,23)(16,25)(18,28)(21,29)(22,27)(24,30)(31,32);(1,3,5,14)(2,7,9,19)(4,12,15,27)(6,13,17,29)(8,22,23,26)(10,21,11,20)(16,28,30,32)(18,25,31,24);(1,4,6,16)(2,8,11,25)(3,12,13,28)(5,15,17,30)(7,18,20,26)(9,23,10,24)(14,27,29,32)(19,31,21,22);(2,9)(7,19)(8,23)(10,11)(18,31)(20,21)(22,26)(24,25)Polycyclic presentation:
2^9|b^2=h,c^2=i,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,f^b=fi,e^c=eh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72238035, #72238036, #72251099, … |
| maximal normal subgroup of | #72238035, #72238036, #72251099, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140741);