Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115163 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C26 | C23 |
| prime radical | C2×#2410 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
2821 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 2821 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,13)(12,14)(15,16);(1,3)(2,5)(4,8)(6,7)(13,16);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)Polycyclic presentation:
2^9|a^2=g,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,g^b=gh,d^c=di,g^c=gh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72210578, #72216625, #72216629, … |
| maximal normal subgroup of | #72210578, #72216625, #72216629, … |
| maximal quotient of | #72210534, #72213435, #72213476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22359);