Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| automorphism | Aut(G) order 25165824 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3686 conjugacy classes of subgroups; only the 802 normal subgroups are shown.
This group has 3686 conjugacy classes of subgroups — too many to draw. Only its 802 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) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,14)(12,16)(13,15)(17,18,20,22)(19,23,24,21)(26,29)(30,31);(2,5)(6,8)(9,11,13,16)(10,14,15,12)(17,18,20,22)(19,21,24,23)(25,26,28,29)(27,31,32,30);(9,11)(10,12)(13,16)(14,15)(17,19)(18,21)(20,24)(22,23)(25,27,28,32)(26,30,29,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,24);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,21)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5862666);