Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13709 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2226 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C23 | #509 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
198 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 94 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)(6,13,16,29)(7,10)(8,20)(9,11)(14,30,31,36)(15,32,26,28)(17,22)(18,19)(21,27,35,25)(23,34)(24,33);(5,6,11,26,12,16,9,15)(7,14,18,27,19,31,10,25)(8,21,23,36,24,35,22,30)(13,17,32,33,29,34,28,20);(1,2,3,4)(5,7,11,18,12,19,9,10)(6,14,26,27,16,31,15,25)(8,17,23,33,24,34,22,20)(13,21,32,36,29,35,28,30)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=ei,d^2=ghi,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dg,f^b=fi,g^b=gh,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72216258, #72216271, #72216273, … |
| maximal normal subgroup of | #72216258, #72216271, #72216273, … |
| maximal quotient of | #72225486 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60069);