Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #102857 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator, Frattini, cosocle | C43 | C23 |
| socle | C23 | #392 |
| prime radical | #23798 | C2 |
| LCS | G ▷ C43 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
609 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 609 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,6)(4,8)(5,7)(10,14)(11,15)(12,16)(17,18)(19,26)(20,24)(21,29)(22,25)(23,27)(28,32)(30,31);(2,6)(3,5)(7,8)(9,10)(11,13)(12,15)(14,16)(17,19)(18,23)(20,27)(21,30)(22,28)(24,32)(25,31)(26,29);(1,2,4,8)(3,6,5,7)(9,11,13,10)(12,14,16,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,26,31,32)Polycyclic presentation:
2^9|c^2=g,d^2=g,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fgi,c^b=cf,d^b=dg,e^b=eghi,f^b=fhi,d^c=dh,e^c=ei,f^c=fhi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72211663, C2×#102860 |
| maximal normal subgroup of | #72211663, C2×#102860 |
| maximal quotient of | #72211633, C2×#102860 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10056);