Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #119683 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1440 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,3,8,6)(9,10,13,18,14,15,12,17)(11,19,20,23,21,16,22,24);(10,15)(11,20)(12,13)(16,23)(19,24)(21,22);(1,3)(2,5)(4,6)(7,8)(9,11)(10,16,15,19)(12,22)(13,20)(14,21)(17,23,18,24)Polycyclic presentation:
2^9|a^2=g,c^2=d,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,f^b=fh,g^b=gh,e^c=ehi,f^c=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72238761, #72238762, #72238776 |
| maximal normal subgroup of | #72238761, #72238762, #72238776 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26886);