Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134692 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1807 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(6,12,15,26)(7,17)(8,20)(9,10)(13,28,30,36)(14,27,16,25)(18,33)(19,23)(21,22)(24,31,35,29);(1,2)(3,4)(5,6,8,12,10,16,20,27,11,15,22,26,9,14,21,25)(7,13,18,24,17,29,32,28,19,30,33,35,23,31,34,36);(2,4)(5,7)(6,13)(8,18)(9,23)(10,17)(11,19)(12,24)(14,31)(15,30)(16,29)(20,32)(21,34)(22,33)(25,36)(26,35)(27,28)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239802, #72239814, #72239817, … |
| maximal normal subgroup of | #72239802, #72239814, #72239817, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55483);