Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6359664 |
| 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 | #61582 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,20)(18,24)(19,23)(21,22);(1,3,5,4)(2,6,8,7)(10,14)(11,18)(12,20)(15,23)(16,17)(19,21);(2,7)(3,4)(6,8);(9,11,13,19)(10,15,16,23)(12,21,20,18)(14,22,17,24);(10,16)(14,17)(15,23)(22,24)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^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=ci,d^b=df,f^b=fh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72339688, #72339719, #72339740 |
| maximal normal subgroup of | #72339688, #72339719, #72339740 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266900);