Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115690 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C26 | C23 |
| prime radical | C2×#2846 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
9165 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 9165 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15);(6,13)(9,15)Polycyclic presentation:
2^9|a^2=g,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,e^b=ei,g^b=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72209716, #72210482, #72210489, … |
| maximal normal subgroup of | #72209716, #72210482, #72210489, … |
| maximal quotient of | #72210553, #72213256, #72213462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22886);