Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #397 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,15,18,11)(8,19)(9,23)(10,13)(12,22)(14,24)(16,30,20,33)(17,26,32,29)(21,34)(25,31,28,35)(27,36);(2,4)(5,7)(6,11)(8,20,21,16)(9,17)(10,18)(12,25,27,28)(13,15)(14,26)(19,33,34,30)(22,35,36,31)(23,29)(24,32);(5,8,9,22,10,21,24,36)(6,12,14,19,13,27,23,34)(7,16,17,31,18,20,32,35)(11,25,26,33,15,28,29,30)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,f^b=fi,g^b=gi,d^c=di,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237652, #72237657, #72237668, … |
| maximal normal subgroup of | #72237652, #72237657, #72237668, … |
| maximal quotient of | #72239693, C2×#123940, #72316889 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31136);