Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| socle | C22 | #1889 |
| prime radical | #537 | C2×C4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
409 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 409 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(2,6,8,11)(3,9,10,12)(17,18,22,27,23,28,35,42)(19,29,33,45,34,46,47,39)(20,25,21,26,30,38,36,43)(24,37,41,48,40,31,44,32);(1,2,4,11,5,8,13,6)(3,7,12,15,10,16,9,14)(17,19,30,37,23,34,20,31)(18,24,38,45,28,40,25,39)(21,32,22,33,36,48,35,47)(26,29,27,41,43,46,42,44);(1,3)(2,7)(4,12)(5,10)(6,14)(8,16)(9,13)(11,15)(17,20,23,30)(18,25,28,38)(19,31,34,37)(21,22,36,35)(24,39,40,45)(26,42,43,27)(29,44,46,41)(32,33,48,47)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,d^2=g,e^2=g,f^2=i,b^a=bd,c^a=ce,e^a=egh,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,e^c=eg,f^c=fh,e^d=ei,f^e=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44470);