Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #486 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2023 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
708 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)(7,16)(8,21)(9,12)(10,15)(11,17)(13,25)(14,24)(18,32)(19,30)(20,33)(22,28)(23,29)(26,36)(27,35)(31,34);(5,7)(6,11,15,27)(8,22)(9,18)(10,20)(12,28,14,26)(13,30,31,32)(16,21,33,29)(17,34)(19,24)(23,36)(25,35);(6,12,15,14)(8,23)(11,26,27,28)(13,21,31,29)(16,32,33,30)(17,35)(22,36)(25,34)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=hi,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,h^b=hi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231475, #72231508, #72231521, … |
| maximal normal subgroup of | #72231475, #72231508, #72231521, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56650);