Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12102 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1852 |
| prime radical | C24 | #107 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
271 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 271 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(2,7)(3,6)(4,11)(8,16)(9,15)(10,13)(12,14)(17,18,21,25,22,26,32,38)(19,23,29,35,28,34,42,46)(20,24,31,37,30,36,43,47)(27,39,41,48,40,33,44,45);(1,2)(3,8)(4,12)(5,9)(6,14)(7,11)(10,15)(13,16)(17,19,21,29,22,28,32,42)(18,23,25,35,26,34,38,46)(20,27,31,41,30,40,43,44)(24,33,37,45,36,39,47,48);(1,3,5,6)(2,8,9,14)(4,10,13,11)(7,12,15,16)(17,20)(18,24)(19,27)(21,31)(22,30)(23,33)(25,37)(26,36)(28,40)(29,41)(32,43)(34,39)(35,45)(38,47)(42,44)(46,48)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dg,e^b=eh,g^b=gi,d^c=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43310);