Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| 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 | |
2109 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,22)(13,16)(14,23)(15,19)(17,21)(18,24);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,23)(13,15)(14,21)(17,18)(19,20)(22,24);(9,12)(10,11)(13,19)(14,16)(15,17)(18,21)(20,24)(22,23);(1,3,5,4)(2,7,8,6)(9,13)(10,17)(11,15)(12,19)(14,22)(16,23)(18,20)(21,24);(1,4,5,3)(2,7,8,6)(9,14,15,24)(10,18,19,23)(11,21,13,22)(12,16,17,20)Polycyclic presentation:
2^9|b^2=fi,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=cgi,d^b=dh,g^b=gi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7429397);