Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500908 |
| 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 | |
1812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,7)(6,8)(9,10,14,24,15,12,16,19)(11,20,22,23,13,17,21,18);(1,3)(2,6)(4,7)(5,8)(9,11,15,13)(10,17,12,20)(14,22,16,21)(18,19,23,24);(9,12,15,10)(11,18,13,23)(14,24,16,19)(17,21,20,22);(2,7)(3,8)(9,13)(10,18)(11,15)(12,23)(14,22)(16,21)(17,24)(19,20);(1,4,5,6)(2,3,7,8)(9,14,15,16)(10,19,12,24)(11,21,13,22)(17,18,20,23)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,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, 7411154);