Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150605 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11801 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | #408 | C23 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #2922 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
226 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,10)(3,9,14,25)(5,12,17,28)(6,11,18,26)(7,20,22,30)(8,16,24,19)(13,23,29,32)(15,21,31,27);(2,7,12,23)(3,13)(4,16)(6,19)(8,9,11,21)(10,25,28,27)(15,30)(17,18)(20,29)(22,24,32,26);(2,8,12,11)(4,17)(7,21,23,9)(10,26,28,24)(14,31)(16,18)(20,29)(22,25,32,27)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=h,d^2=g,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=dg,e^b=efgi,g^b=gi,e^c=ei,e^d=eh,f^e=fi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57801);