Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357223 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2628 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)(3,6)(7,8)(10,14)(12,19)(15,20)(16,21)(18,23)(22,24);(9,10,12,14,13,15,19,20)(11,16,18,21,17,22,23,24);(1,2,6,8,4,7,3,5)(10,15)(14,20)(16,22)(21,24);(9,11)(10,16)(12,18)(13,17)(14,21)(15,22)(19,23)(20,24);(11,17)(16,22)(18,23)(21,24)Polycyclic presentation:
2^9|b^2=f,c^2=gi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,e^d=eh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72214074, #72214466, #72215816, … |
| maximal normal subgroup of | #72214074, #72214466, #72215816, … |
| maximal quotient of | #72224764 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264426);