Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6360896 |
| 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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8)(3,6,4,7)(9,10,16,20)(11,21,12,13)(14,19,15,18)(17,23,24,22);(1,3,5,4)(2,6,8,7)(9,11)(10,17)(12,16)(13,19)(14,22)(15,23)(18,21)(20,24);(2,7)(3,4)(6,8)(9,12,16,11)(10,13,20,21)(14,22,15,23)(17,18,24,19);(9,13)(10,11)(12,20)(14,17)(15,24)(16,21)(18,22)(19,23);(9,14,16,15)(10,18,20,19)(11,22,12,23)(13,17,21,24)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,e^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=chi,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72338427, #72338436, #72338444 |
| maximal normal subgroup of | #72338427, #72338436, #72338444 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6268125);