Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392239 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14681 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3545 conjugacy classes of subgroups; only the 432 normal subgroups are shown.
This group has 3545 conjugacy classes of subgroups — too many to draw. Only its 432 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)(3,11,14,7)(4,15)(5,9)(6,10)(8,17)(12,19,26,23)(13,24,27,20)(16,21)(18,22)(25,31,32,30)(28,29);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,20)(10,11)(15,23)(16,25)(17,26)(18,27)(21,30)(22,24)(28,31)(29,32);(1,4)(2,8,10,15)(3,12)(5,16)(6,17)(7,19,11,23)(9,21,22,28)(13,25)(14,26)(18,29)(20,30,24,31)(27,32);(1,5)(2,9)(3,13)(4,16)(6,18)(7,20)(8,21)(10,22)(11,24)(12,25)(14,27)(15,28)(17,29)(19,30)(23,31)(26,32)Polycyclic presentation:
2^9|a^2=h,c^2=e,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^b=hi,f^c=fi,g^c=gi,f^d=fi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72237917, #72237919, #72240469, … |
| maximal normal subgroup of | #72237917, #72237919, #72240469, … |
| maximal quotient of | #72265195, #72265917 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299435);