Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142308 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1598 |
| commutator, Frattini, cosocle | C22×M16 | C23 |
| prime radical | C2×#2190 | C2 |
| LCS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
609 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 609 conjugacy classes of subgroups — too many to draw. Only its 70 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:
(1,2)(3,4)(6,10)(7,15)(8,17)(11,13)(12,16)(14,19)(18,20);(5,6)(7,16)(8,12)(9,14)(10,15)(11,18)(13,17)(19,20);(2,4)(5,7,8,15,9,18,17,20)(6,11,13,10,14,16,12,19)Polycyclic presentation:
2^9|c^2=g,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,g^b=gi,h^b=hi,d^c=dh,f^c=fi,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72250258, #72250333, #72250339, … |
| maximal normal subgroup of | #72250258, #72250333, #72250339, … |
| maximal quotient of | #72264958, #72264968, #72307653 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49504);