Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6361570 |
| 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 | |
1897 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)(3,6)(4,7)(5,8)(10,14)(12,20)(15,21)(16,17)(18,24)(22,23);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,16)(13,17)(14,20)(18,22)(19,23)(21,24);(2,7)(3,4)(6,8)(9,11,13,19)(10,15,17,23)(12,18,20,24)(14,21,16,22);(2,8)(6,7)(10,16)(12,20)(14,17)(15,22)(18,24)(21,23);(10,17)(14,16)(15,23)(21,22)Polycyclic presentation:
2^9|b^2=i,c^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6268792);