Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144639 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10390 |
| commutator, Frattini, cosocle | C23×D4 | C23 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
544 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 544 conjugacy classes of subgroups — too many to draw. Only its 46 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)(3,6)(4,12,14,8)(5,9)(7,10,16,13)(11,15);(1,3)(2,6)(4,13,14,10)(5,11)(7,12,16,8)(9,15);(1,3)(2,7)(4,10)(5,11)(6,8)(9,16)(12,15)(13,14)Polycyclic presentation:
2^9|a^2=d,b^2=d,b^a=bd,c^a=ce,e^a=eg,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fg,h^b=hi,d^c=dg,h^c=hi,e^d=ei,f^e=fi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72216336, #72216344, #72216381, … |
| maximal normal subgroup of | #72216336, #72216344, #72216381, … |
| maximal quotient of | #72241344, #72252379, C2×#144639, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51835);