Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150123 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | #33713 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
334 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,8,13,19,29,35,27,10,15,23,26,9,14,22,28,11,16,7,17,20,31,36,33,24,30,21,34,25,32,18,12);(6,12,16,28)(7,18)(8,22)(9,20)(10,24)(13,14,17,32)(15,33,30,27)(19,25)(21,36)(23,35)(26,29,34,31);(1,2,3,4)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=e,d^2=ghi,f^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,f^b=fg,g^b=gh,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72225478, #72225508, #72225531, … |
| maximal normal subgroup of | #72225478, #72225508, #72225531, … |
| maximal quotient of | #72225471, #72225478 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60188);