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 | |
1795 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)(4,8)(5,6)(9,10,14,19)(11,21,23,16)(12,22,17,24)(13,20,15,18);(2,6)(7,8)(9,11,14,23)(10,16,19,21)(12,18,17,20)(13,22,15,24);(1,3,5,4)(2,7,6,8)(9,12,14,17)(10,11,19,23)(13,16,15,21)(18,22,20,24);(9,11)(10,17)(12,19)(13,22)(14,23)(15,24)(16,20)(18,21);(1,4,5,3)(2,7,6,8)(9,13,14,15)(10,18,19,20)(11,22,23,24)(12,16,17,21)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,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, 7426902);