Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(14,19)(16,21)(20,23)(22,28);(5,7,9,8)(6,10,12,11)(13,14,17,19)(15,22,24,28)(16,26,21,18)(20,27,23,25);(2,4)(5,7,9,8)(6,11,12,10)(13,15,17,24)(14,20,19,23)(16,22,21,28)(18,27,26,25);(5,7,9,8)(6,11,12,10)(13,16,17,21)(14,18,19,26)(15,23,24,20)(22,27,28,25);(5,8,9,7)(6,10,12,11)(14,21)(16,19)(18,26)(20,28)(22,23)(25,27)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72249945, #72256983, #72256988, … |
| maximal normal subgroup of | #72249945, #72256983, #72256988, … |
| maximal quotient of | #72265448, #72307605, #72307861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486275);