Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10382 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1446 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
651 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)(11,18)(12,21)(13,17)(14,19)(15,20)(16,22)(23,24);(5,7)(6,8)(9,11)(10,14,17,18)(12,20)(13,19)(15,21,23,16)(22,24);(1,3)(2,4)(5,6)(7,8)(9,11)(10,15,17,23)(12,20)(13,19)(14,16,18,21)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,e^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi,e^c=eh,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29368);