Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13569 |
| 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×Q8 | C2×C42 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | C2×#454 | C22 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
492 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 492 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) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,16)(8,20)(9,13)(10,15)(11,18)(12,22)(14,26)(17,33)(19,34)(21,28)(23,31)(24,30)(25,32)(27,35)(29,36);(5,7)(6,11,13,28)(8,21,22,36)(9,24,26,19)(10,17)(12,30,31,33)(14,29,15,27)(16,20,34,32)(18,23)(25,35);(1,3,4,2)(7,17)(11,27)(16,33)(18,35)(19,24)(21,36)(28,29)(30,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=i,g^2=i,b^a=bd,d^a=di,f^a=fg,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59108);