Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125098 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
414 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 414 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2)(3,4)(6,9,10,11)(13,14,18,25,20,26,19,24)(15,27,30,41,32,43,31,37)(16,23,17,22,35,36,34,42)(21,28,39,44,40,33,38,29);(5,6,7,11,8,10,12,9)(13,15,17,29,18,30,35,28,20,32,34,44,19,31,16,33)(14,21,36,43,24,38,22,41,26,40,23,27,25,39,42,37);(2,4)(13,16,19,34,20,35,18,17)(14,22,25,36,26,42,24,23)(15,28,31,29,32,33,30,44)(21,37,39,27,40,41,38,43)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=g,d^2=ghi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ci,d^b=dh,f^b=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45222);