Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6358586 |
| 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 | C42 | C25 |
| prime radical | C2×#3274 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1905 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,7)(3,5,8,6)(9,10)(11,17)(12,21)(13,15)(14,20)(16,22)(18,24)(19,23);(9,11,13,18)(10,14,15,23)(12,19,22,20)(16,24,21,17);(2,5)(3,8)(6,7)(11,18)(14,23)(17,24)(19,20);(11,19)(12,22)(14,24)(16,21)(17,23)(18,20);(10,15)(14,23)(16,21)(17,24)Polycyclic presentation:
2^9|a^2=i,b^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,f^b=fh,g^c=gi,f^d=fh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72339688, #72339719, #72339747 |
| maximal normal subgroup of | #72339688, #72339719, #72339747 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265790);