Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13697 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D16 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
244 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(14,19)(15,20)(16,25)(17,28)(21,24)(22,39)(23,38)(26,43)(27,36)(29,42)(30,37)(31,32)(33,40)(34,35)(41,44);(2,4)(6,9)(10,11)(13,14,20,32,16,22,34,38,28,40,43,44,42,30,27,21,18,24,36,37,29,41,26,33,17,23,35,39,25,31,15,19);(5,7,8,12)(6,10,9,11)(13,15,25,35,17,26,29,36,18,27,42,43,28,34,16,20)(14,21,31,44,23,38,41,32,24,19,30,39,40,33,22,37)Polycyclic presentation:
2^9|a^2=f,b^2=def,c^2=eh,d^2=gh,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,h^a=hi,c^b=ci92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60213);