Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33388 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#102 |
| commutator | C8 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C2 | C22×D4⋊C8 |
| prime radical | #3584 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#131 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
989 conjugacy classes of subgroups; only the 352 normal subgroups are shown.
This group has 989 conjugacy classes of subgroups — too many to draw. Only its 352 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:
(1,2,6,11,7,12,23,22,8,13,5,16,21,3,14,10)(4,9,19,26,20,27,32,31,17,24,18,29,30,15,28,25);(1,3)(2,7)(4,15)(5,10)(6,22)(8,12)(9,20)(11,14)(13,21)(16,23)(17,27)(18,25)(19,31)(24,30)(26,28)(29,32);(1,4)(2,9)(3,15)(5,18)(6,19)(7,20)(8,17)(10,25)(11,26)(12,27)(13,24)(14,28)(16,29)(21,30)(22,31)(23,32);(4,17)(9,24)(15,27)(18,19)(20,30)(25,31)(26,29)(28,32)Polycyclic presentation:
2^9|a^2=f,e^2=g,f^2=h,g^2=i,h^2=i,b^a=be,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72215816, #72219677, #72219678, … |
| maximal normal subgroup of | #72215816, #72219677, #72219678, … |
| maximal quotient of | #72286692, D4×#1336, C2×#481073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388269);