Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136643 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,16,8,14,11,26)(7,17,18,28,19,12,21,29)(9,23,24,31,22,13,25,15)(20,30,33,27,34,36,35,32);(1,2,3,4)(5,7,8,19)(6,12,14,17)(9,20,22,34)(10,18,11,21)(13,30,23,36)(15,27,31,32)(16,29,26,28)(24,33,25,35);(6,13)(7,18)(9,24)(10,11)(12,27)(14,23)(15,26)(16,31)(17,32)(19,21)(20,34)(22,25)(28,36)(29,30)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,f^b=fi,e^c=ehi,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250548, #72250560, #72250569, … |
| maximal normal subgroup of | #72250548, #72250560, #72250569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55246);