Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #486 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2041 |
| 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:
(1,2,3,4)(5,6)(7,16)(8,22)(9,18)(10,13)(11,15)(12,17)(14,24)(19,32)(20,29)(21,30)(23,33)(25,28)(26,31)(27,36)(34,35);(5,7,9,10,12,21,11,20)(6,13,15,16,17,29,18,30)(8,19,23,26,25,24,27,35)(14,22,31,36,32,28,34,33);(5,8,9,23,12,25,11,27)(6,14,15,31,17,32,18,34)(7,19,10,26,21,24,20,35)(13,28,16,33,29,22,30,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=hi,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=ehi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249862, #72249872, #72249900, … |
| maximal normal subgroup of | #72249862, #72249872, #72249900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55923);