Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13334 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
728 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)(5,6)(7,15)(8,20)(9,25)(10,13)(11,18)(12,23)(14,22)(16,32)(17,31)(19,33)(21,30)(24,34)(26,28)(27,35)(29,36);(5,6)(7,16,17,11)(8,12)(9,14)(10,13)(15,18,31,32)(19,30,26,27)(20,23)(21,33,35,28)(22,25)(24,29)(34,36);(5,7)(6,11)(8,21)(9,26)(10,17)(12,28)(13,16)(14,30)(15,20)(18,23)(19,22)(24,35)(25,27)(29,33)(31,34)(32,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,f^b=fi,d^c=di,e^c=eh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320262, #72320289, #72320421, … |
| maximal normal subgroup of | #72320262, #72320289, #72320421, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55258);