Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514148 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1749 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,22)(12,24)(13,21)(14,20)(15,19)(16,17)(18,23);(1,3,5,4)(2,6,8,7)(9,11,14,17)(10,16,20,22)(12,19,18,21)(13,23,15,24);(9,12)(10,17)(11,20)(13,16)(14,18)(15,22)(19,24)(21,23);(1,3,5,4)(2,7,8,6)(9,11)(10,18)(12,20)(13,23)(14,17)(15,24)(16,21)(19,22);(1,4,5,3)(2,7,8,6)(9,13,14,15)(10,19,20,21)(11,23,17,24)(12,16,18,22)Polycyclic presentation:
2^9|a^2=fi,b^2=f,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,g^c=gi,h^c=hi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7430000);