Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13402 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2034 |
| 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 | |
698 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)(5,6,11,18,13,19,12,29)(7,20,23,36,24,33,28,32)(8,17,10,27,25,14,9,16)(15,30,34,21,35,22,31,26);(6,14)(7,21)(8,10)(9,25)(11,12)(15,31)(16,29)(17,19)(18,27)(20,33)(22,28)(23,30)(24,26)(34,35);(2,4)(5,7,8,22,12,23,9,26,13,24,25,30,11,28,10,21)(6,15,16,32,18,31,17,33,19,35,27,36,29,34,14,20)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=h,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250238, #72250248, #72250282, … |
| maximal normal subgroup of | #72250238, #72250248, #72250282, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55753);