Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1762 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,8,12,11,16,10,14)(7,17,20,33,21,34,18,31)(9,23,22,32,24,30,26,15)(13,25,27,36,28,35,29,19);(1,2,4,3)(6,12,16,14)(9,24)(13,27,28,29)(15,30,32,23)(17,31,34,33)(19,36)(22,26)(25,35);(1,3,4,2)(5,7)(6,13)(8,20)(9,25)(10,18)(11,21)(12,27)(14,29)(15,31)(16,28)(17,23)(19,26)(22,36)(24,35)(30,34)(32,33)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=dh,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,g^a=gi,h^a=hi,e^b=ei,d^c=di,e^c=eh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264495, #72264503, #72264525, … |
| maximal normal subgroup of | #72264495, #72264503, #72264525, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54105);