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 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2452 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 2452 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)(3,8)(4,16)(5,10)(6,19)(7,11)(9,17)(12,27)(13,23)(14,22)(15,24)(18,25)(20,26)(21,28)(29,31)(30,32);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,21)(10,22)(11,24)(16,31)(17,28)(18,29)(19,23)(20,30)(25,27)(26,32);(1,4)(2,9)(3,12)(5,17)(6,20)(7,18)(8,21)(10,25)(11,26)(13,28)(14,30)(15,29)(16,19)(22,27)(23,31)(24,32);(1,5,7,6)(2,10,11,19)(3,13,15,14)(4,17,18,20)(8,22,24,23)(9,25,26,16)(12,28,29,30)(21,27,32,31)Polycyclic presentation:
2^9|d^2=i,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 | #72233206, #72238342, #72249340, … |
| maximal normal subgroup of | #72233206, #72238342, #72249340, … |
| maximal quotient of | #72233087, #72234415, #72249366, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419005);