Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6443048 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60639 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2266 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)(3,8)(4,6)(5,7)(9,10,13,17,14,19,24,18)(11,12,20,22,16,23,21,15);(3,4)(6,8)(9,11)(10,15)(12,18)(13,21)(14,16)(17,23)(19,22)(20,24);(1,3,5,4)(2,6,7,8)(9,12,13,22,14,23,24,15)(10,16,18,20,19,11,17,21);(1,4,5,3)(2,6,7,8)(9,13,14,24)(10,17,19,18)(11,20,16,21)(12,22,23,15);(9,13,14,24)(10,17,19,18)(11,21,16,20)(12,15,23,22)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370228);