Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9109 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C24 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C2×#525 | C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1044 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)(5,6,8,13,9,14,19,26,10,11,18,22,20,23,30,33)(7,12,16,24,17,25,29,34,15,21,27,31,28,32,35,36);(6,11)(8,18)(14,23)(16,27)(19,30)(24,31)(29,35)(34,36);(2,4)(5,7)(6,12)(8,16)(9,17)(10,15)(11,21)(13,24)(14,25)(18,27)(19,29)(20,28)(22,31)(23,32)(26,34)(30,35)(33,36)Polycyclic presentation:
2^9|a^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,g^b=gi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72281309, #72281319, #72281343, … |
| maximal normal subgroup of | #72281309, #72281319, #72281343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54737);