Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #507998 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #32930 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
777 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 777 conjugacy classes of subgroups — too many to draw. Only its 154 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 — 4 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,18,21,25)(19,22,28,31)(20,29,27,24)(23,32,30,26);(1,3)(2,6)(4,7)(5,8)(10,14)(13,16)(18,22)(19,26)(20,27)(23,24)(25,31)(28,32)(29,30);(3,7)(6,8)Polycyclic presentation:
2^9|b^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72343796, #72343851, #72343853, … |
| maximal normal subgroup of | #72343796, #72343851, #72343853, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415676);