Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10078 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C2×#2819 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10,13,14,12,11,15,16)(18,21,22,23);(5,6)(7,8)(13,15)(14,16)(17,18)(19,23)(20,22)(21,24);(2,4)(6,8)(9,11)(10,12)(13,14)(15,16)(19,24)(21,23)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,e^b=ehi,g^b=ghi,d^c=dhi,e^c=eh,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72217670 |
| maximal normal subgroup of | #72217670 |
| maximal quotient of | #72216328 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18445);