Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13561 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×D4 | C2×C42 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | C2×#453 | C22 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
544 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 544 conjugacy classes of subgroups — too many to draw. Only its 86 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:
(1,2,4,3)(5,6)(7,13)(8,17)(9,12)(10,15)(11,18)(14,20)(16,19);(5,7)(6,10,11,13)(8,16,18,15)(9,14)(12,19,17,20);(1,3,4,2)(7,14)(10,19)(13,20)(15,16)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,g^2=i,b^a=bd,f^a=fgi,g^a=gi,h^a=hi,c^b=ci,d^b=dg,g^b=gh,h^b=hi,f^d=fhi,g^d=gi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279279, #72279448 |
| maximal normal subgroup of | #72279279, #72279448 |
| maximal quotient of | #72279731 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59102);