Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 12582912 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C23×C44 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15947 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3314 conjugacy classes of subgroups; only the 490 normal subgroups are shown.
This group has 3314 conjugacy classes of subgroups — too many to draw. Only its 490 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:
(1,2)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,21)(22,24)(26,28)(27,31);(2,3)(6,7)(18,20)(19,24)(26,28)(30,32);(4,8)(6,7)(18,20)(22,23)(25,26,29,28)(27,30,31,32);(1,3,5,2)(4,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,24)(18,22,20,23)(25,27,29,31)(26,30,28,32);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,13,14,16)(17,20,21,18)(19,22,24,23)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=fhi,c^2=h,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72327843, #72327881, #72327930 |
| maximal normal subgroup of | #72327843, #72327881, #72327930 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4028742);