Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12503 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1957 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
335 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 335 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,7,6,10,5,9)(3,11,12,16,14,15,13,8)(17,18)(19,27)(20,32)(21,24)(22,35)(23,29)(25,28)(26,33)(30,39)(31,46)(34,47)(36,45)(37,48)(38,43)(40,42)(41,44);(2,7,10,9)(8,11,16,15)(17,19,28,42,22,31,34,45,21,30,43,48,33,44,20,29)(18,23,32,41,26,37,38,39,24,36,47,46,35,40,25,27);(1,3,5,13,6,14,4,12)(2,8,7,11,10,16,9,15)(18,24)(23,36)(25,38)(26,35)(27,39)(32,47)(37,40)(41,46)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=ei,d^2=gi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49010);