Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13215 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1961 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(6,12)(8,22)(9,24)(10,23)(13,30)(14,28)(15,16)(17,32)(18,19)(20,35)(25,36)(26,31)(27,29)(33,34);(5,6,9,15,10,16,8,12,11,14,22,29,23,27,24,28)(7,17,19,30,20,31,36,33,21,34,25,26,35,13,18,32);(5,7)(6,13,14,31)(8,18)(9,25)(10,20)(11,21)(12,26,28,30)(15,33,29,32)(16,17,27,34)(19,22)(23,35)(24,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,d^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,c^b=ch,e^b=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240739, #72249886, #72249947, … |
| maximal normal subgroup of | #72240739, #72249886, #72249947, … |
| maximal quotient of | #72240742 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54200);