Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C23×H(5,𝔽2) | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
17806 conjugacy classes of subgroups; only the 706 normal subgroups are shown.
This group has 17806 conjugacy classes of subgroups — too many to draw. Only its 706 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,15,16,13)(12,14);(9,11)(10,13)(12,16)(14,15);(2,4)(9,10)(11,13)(12,14)(15,16);(6,8)Polycyclic presentation:
2^9|a^2=h,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^b=fi,h^b=hi,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72219111, #72219118, #72225609, … |
| maximal normal subgroup of | #72219118, #72225609, #72225691, … |
| maximal quotient of | #72218467, #72218475, #72218980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299321);