Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1999 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1124 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)(7,15)(8,19)(9,24)(10,13)(11,16)(12,22)(14,21)(17,31)(18,33)(20,30)(23,34)(25,28)(26,32)(27,35)(29,36);(5,7)(6,11)(8,20)(9,25)(10,17)(12,28)(13,26)(14,30)(15,19)(16,22)(18,21)(23,35)(24,27)(29,33)(31,34)(32,36);(11,26)(15,31)(27,30)(28,33)Polycyclic presentation:
2^9|a^2=g,d^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=ei,h^b=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231488, #72231499, #72231508, … |
| maximal normal subgroup of | #72231488, #72231499, #72231508, … |
| maximal quotient of | #72265025 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54838);