Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13459 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #498 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2058 |
| 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 | |
542 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)(7,17)(8,20,23,30)(9,25)(10,12)(11,16)(13,22,32,18)(14,26)(15,19)(21,34,24,35)(27,33,36,31)(28,29);(1,2,4,3)(5,7,9,10,11,19,26,28)(6,12,14,17,16,29,25,15)(8,18,21,27,23,22,24,36)(13,30,31,34,32,20,33,35);(1,3,4,2)(5,8,9,21,11,23,26,24)(6,13,14,31,16,32,25,33)(7,18,10,27,19,22,28,36)(12,20,17,35,29,30,15,34)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,e^c=eh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249871, #72249876, #72249900, … |
| maximal normal subgroup of | #72249871, #72249876, #72249900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55918);