Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7076 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #327 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C22⋊C8 | C23 |
| socle | C22 | C2×#327 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
358 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 358 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2)(3,7)(4,11,12,6)(5,8)(9,15,16,13)(10,14)(17,18)(19,28,30,45)(20,33,35,44)(21,26)(22,38)(23,27)(24,29)(25,36,42,37)(31,41,32,48)(34,43)(39,46)(40,47);(2,6)(7,13)(8,11)(14,15)(17,19,23,30)(18,24,27,40)(20,34,35,46)(21,32,22,31)(25,41,42,48)(26,43,38,39)(28,44,45,33)(29,36,47,37);(1,3,5,10)(2,7,8,14)(4,9,12,16)(6,13,11,15)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=h,d^2=gi,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29735);