Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94184 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1390 |
| commutator | C2×C4⋊C4 | C42 |
| Frattini, cosocle | C4×#124 | C22 |
| prime radical | C22×D4 | C4⋊C4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C4×#124 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
216 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 64 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2,4,9,6,10,5,8)(3,11,12,7,14,15,13,16)(17,18)(19,27,29,43)(20,32,33,42)(21,35,37,26)(22,25)(23,28,38,46)(24,34)(30,45,31,44)(36,41)(39,47,40,48);(1,3,5,13,6,14,4,12)(2,7,9,15,10,16,8,11)(17,19)(18,23,25,39)(20,28)(21,29)(22,31)(24,27,41,44)(26,38,35,40)(30,37)(32,43,42,45)(33,47)(34,48)(36,46)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fg,d^2=fgi,e^2=fgi,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gh,h^a=hi,d^b=dg,f^b=fi,g^b=gi,h^b=hi,d^c=dhi,f^c=fi,g^c=gi,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1456);