Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #112521 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C2×#2867 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
2344 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 2344 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) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,6,4,5,7,8)(9,10)(11,13)(12,15)(14,16);(2,5)(3,7)(10,13)(11,16)(14,15);(2,5)(3,7)(11,16)(13,14)Polycyclic presentation:
2^9|a^2=g,d^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,g^b=ghi,d^c=di,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214390, #72214533, #72219741, … |
| maximal normal subgroup of | #72214390, #72219741, #72219742, … |
| maximal quotient of | #72228190, #72228196 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 19725);