Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1440 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1434 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)(5,6,10,14,11,15,9,26)(7,16,19,29,20,28,18,12)(8,21,24,32,25,31,23,13)(17,27,33,30,34,35,22,36);(1,2)(3,4)(5,7)(6,12,15,29)(8,22)(9,18)(10,19)(11,20)(13,30,32,36)(14,16,26,28)(17,24)(21,35,31,27)(23,34)(25,33);(1,3,4,2)(5,8,9,23,11,25,10,24)(6,13,14,21,15,32,26,31)(7,17,18,22,20,34,19,33)(12,27,16,30,29,35,28,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265350, #72265363, #72265392, … |
| maximal normal subgroup of | #72265350, #72265363, #72265392, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31042);