Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141346 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1957 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
337 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6)(3,7)(4,10)(8,12)(9,16)(11,15)(13,14)(18,24,28,32)(19,30)(20,25)(21,22)(26,27,31,29);(1,2,7,14,4,9,11,12,5,8,15,16,10,13,3,6)(17,18,25,26,21,27,20,32,23,28,19,31,22,29,30,24);(2,8)(6,12)(9,13)(14,16)(17,19,21,30,23,25,22,20)(18,26,27,32,28,31,29,24)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=e,d^2=ghi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,d^b=dh,f^b=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48542);