Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33531 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(6,12)(7,16)(8,19)(9,23)(10,22)(14,29)(15,24)(17,21)(18,36)(20,26)(25,27)(28,32)(31,35)(33,34);(1,3,4,2)(5,6)(7,13)(8,20)(9,24)(10,26)(11,14)(12,19)(15,22)(16,35)(17,34)(18,30)(21,32)(23,29)(25,31)(27,33)(28,36);(5,7)(6,13)(8,21)(9,25)(10,27)(11,18)(12,28)(14,30)(15,34)(16,23)(17,22)(19,36)(20,32)(24,31)(26,33)(29,35);(6,14)(12,29)(13,30)(15,26)(20,24)(28,35)(31,32)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,d^b=di,e^b=eh,f^b=fhi,h^b=hi,e^c=ehi,f^c=fh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72322465, #72322481, #72322499, … |
| maximal normal subgroup of | #72322465, #72322481, #72322499, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401503);