Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23895 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14127 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2196 conjugacy classes of subgroups; only the 344 normal subgroups are shown.
This group has 2196 conjugacy classes of subgroups — too many to draw. Only its 344 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:
(1,2,5,10,6,9,16,22)(3,7,13,19,12,17,26,28)(4,8,15,21,14,20,27,31)(11,23,25,32,24,18,29,30);(7,17)(18,23)(19,28)(30,32);(1,3,5,13,6,12,16,26)(2,7,10,19,9,17,22,28)(4,11,15,25,14,24,27,29)(8,18,21,30,20,23,31,32);(1,4)(2,8)(3,11)(5,15)(6,14)(7,18)(9,20)(10,21)(12,24)(13,25)(16,27)(17,23)(19,30)(22,31)(26,29)(28,32)Polycyclic presentation:
2^9|a^2=h,c^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,e^c=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240451, #72240455, #72240459, … |
| maximal normal subgroup of | #72240451, #72240455, #72240459, … |
| maximal quotient of | #72256157, C2×#373639 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280835);