Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6131635 |
| automorphism | Aut(G) order 22020096 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C27 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
9318 conjugacy classes of subgroups; only the 546 normal subgroups are shown.
This group has 9318 conjugacy classes of subgroups — too many to draw. Only its 546 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) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10,12,14)(11,15,16,13)(18,21)(19,23)(26,29)(30,32);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(19,23)(22,24)(26,29)(27,31);(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,29)(31,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,21)(23,24)(25,27)(26,30)(28,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=eg68 conjugacy classes — view the full character table.
| Fitting subgroup of | #72227882 |
| maximal subgroup of | #72227882, #72334160, #72334186, … |
| maximal normal subgroup of | #72227882, #72334160, #72334186, … |
| maximal quotient of | #72227854 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6038832);