Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18746 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #60946 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1299 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(10,14,16,19)(11,17,18,20);(9,10,12,19,13,16,21,14)(11,15,20,23,18,24,17,22);(2,5)(3,8)(6,7)(9,11)(10,15)(12,20)(13,18)(14,22)(16,24)(17,21)(19,23);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fi,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fhi,g^c=gh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72342638, #72342646, #72342650 |
| maximal normal subgroup of | #72342638, #72342646, #72342650 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102686);