Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #301861 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1581 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 1581 conjugacy classes of subgroups — too many to draw. Only its 182 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:
(2,5)(3,6)(7,8)(9,10)(11,13)(12,14)(15,16);(1,2,6,8,4,7,3,5)(9,11)(10,13)(12,16)(14,15);(9,10)(11,15,16,13)(12,14);(1,2,6,8,4,7,3,5)Polycyclic presentation:
2^9|b^2=eh,c^2=f,d^2=eh,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,g^c=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72258119, #72268295, D8×B(4,𝔽2), … |
| maximal normal subgroup of | #72258119, #72268295, D8×B(4,𝔽2), … |
| maximal quotient of | #72258178, C2×#301866, #72313570 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 209062);