Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129437 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1084 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1084 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,12)(8,16)(9,13,18,19)(10,14)(11,15,20,17);(1,2)(3,4)(5,7)(6,11)(8,10)(9,17)(12,19)(13,16)(14,20)(15,18);(2,4)(5,8,10,7)(6,12)(9,17,18,15)(11,19)(13,20)(14,16)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,e^c=ei,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72237806, #72256330, #72262193, … |
| maximal normal subgroup of | #72237806, #72256330, #72262193, … |
| maximal quotient of | #72237808, #72309478 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54806);