Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #107658 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1017 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,6,4,3,7,8)(9,10)(11,17)(12,20,21,15)(13,16)(14,19)(18,22,24,23);(5,7)(6,8)(9,11)(10,14,16,17)(12,18)(13,19)(15,22,20,23)(21,24);(1,3)(2,4)(5,6)(7,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 14858);