Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12310 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #453 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
358 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 358 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(6,11)(8,19)(9,22)(12,28)(13,20)(14,26)(15,27)(16,21)(17,24)(23,32)(25,29)(30,36)(31,35)(33,34);(1,3,4,2)(5,6)(7,12)(8,20)(9,13)(10,15)(11,19)(14,22)(16,25)(17,29)(18,31)(21,28)(23,26)(24,30)(27,32)(33,36)(34,35);(5,7,10,18)(6,12,15,31)(8,21,23,34)(9,24,22,17)(11,25,27,36)(13,30,14,29)(16,32,33,19)(20,28,26,35)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=eg,g^b=gi,h^b=hi,d^c=dg,e^c=eh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43685);