Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140598 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1961 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,8,14,11,16,10,27)(7,17,20,12,21,31,18,30)(9,23,22,28,26,36,25,15)(13,24,33,35,29,34,32,19);(1,2,3,4)(5,7)(6,12,16,30)(8,18)(9,19)(10,20)(11,21)(13,15,29,28)(14,17,27,31)(22,34)(23,33,36,32)(24,25)(26,35);(5,7)(6,13)(8,20)(9,24)(10,18)(11,21)(12,28)(14,33)(15,30)(16,29)(17,23)(19,25)(22,35)(26,34)(27,32)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=dh,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,g^b=gi,d^c=di,e^c=eh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320962, #72321051, #72321108, … |
| maximal normal subgroup of | #72320962, #72321051, #72321108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55272);