Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129554 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C24 | #102 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
374 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 106 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)(4,5)(7,8)(9,10,14,15,12,11,13,16)(17,18,19,22,21,26,27,25)(20,29,28,24,31,23,30,32);(2,5)(6,8)(10,11)(15,16)(17,19,21,27)(18,22,26,25)(20,28,31,30)(23,32,29,24);(1,3,4,7)(2,6,5,8)(9,11,12,10)(13,15,14,16)(18,23)(19,27)(20,30)(22,24)(25,32)(26,29)(28,31)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=d,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ch,e^c=egi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72320433, #72320446, #72320561, … |
| maximal normal subgroup of | #72320433, #72320446, #72320561, … |
| maximal quotient of | #72321273, C2×#135284 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42480);