Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357798 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23);(9,11,13,21,14,19,12,20)(10,15,16,18,17,23,22,24);(1,2,3,5,4,6,7,8)(9,10)(11,15)(12,22)(13,16)(14,17)(18,21)(19,23)(20,24);(2,6)(5,8)(11,19)(15,23)(18,24)(20,21);(9,12,14,13)(10,16,17,22)(11,20,19,21)(15,18,23,24)Polycyclic presentation:
2^9|b^2=f,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dh,d^c=di,e^c=eh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214074, #72214114, #72214208, … |
| maximal normal subgroup of | #72214074, #72214114, #72214208, … |
| maximal quotient of | #72216602, #72224773, #72224774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264996);