Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511807 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32898 |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2450 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 2450 conjugacy classes of subgroups — too many to draw. Only its 186 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)(4,9)(5,8)(6,11)(10,18)(12,26)(13,19)(14,22)(15,24)(16,23)(17,21)(20,27)(25,29)(28,32)(30,31);(1,3)(2,7)(4,12)(5,17)(6,14)(8,19)(9,20)(10,21)(11,22)(13,18)(15,27)(16,30)(23,31)(24,26)(25,32)(28,29);(2,8)(5,18)(7,19)(9,23)(10,11)(13,17)(16,29)(20,31)(21,22)(24,25)(26,32)(28,30);(1,4)(2,9)(3,12)(5,16)(6,15)(7,20)(8,23)(10,25)(11,24)(13,28)(14,27)(17,30)(18,29)(19,31)(21,32)(22,26)Polycyclic presentation:
2^9|f^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,h^a=hi,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,e^d=ei,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72234209, #72240648, #72249340, … |
| maximal normal subgroup of | #72234209, #72240648, #72249340, … |
| maximal quotient of | #72249363, #72249364, #72249406, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419003);