Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#535 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1461 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,4,3,6,8)(9,10,14,17,15,11,13,18)(12,19,20,23,22,16,21,24);(1,3,5,8,4,2,6,7)(9,11,15,10)(12,16,22,19)(13,18,14,17)(20,23,21,24);(2,3)(7,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24);(9,13,15,14)(10,17,11,18)(12,20,22,21)(16,23,19,24)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,c^b=ci,e^b=eh,f^b=fh,g^b=gh,e^c=eh,g^c=gh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72280243, #72302216 |
| maximal normal subgroup of | #72280243, #72302216 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100997);