Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13033 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(6,12)(7,17)(8,20)(9,10)(13,29)(14,16)(15,21)(18,36)(19,35)(22,26)(23,24)(25,28)(27,31)(30,34);(5,6,9,15,11,16,10,25)(7,13,18,32,19,29,26,33)(8,21,23,12,24,28,20,14)(17,34,22,31,35,27,36,30);(5,7)(6,13)(8,22)(9,18)(10,26)(11,19)(12,27)(14,34)(15,32)(16,29)(17,20)(21,31)(23,35)(24,36)(25,33)(28,30)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=dhi,e^b=eh,h^b=hi,d^c=dh,e^c=eh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72321294, #72321366, #72321394, … |
| maximal normal subgroup of | #72321294, #72321366, #72321394, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53970);