Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19254 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #60415 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,6)(7,8)(9,10,13,16)(11,14)(12,17,21,22)(15,20)(18,23)(19,24);(1,2,6,8,4,7,3,5)(9,11)(10,14,16,23)(12,20)(13,18)(15,22,24,17)(19,21);(9,11)(10,15)(12,20)(13,18)(14,22)(16,24)(17,23)(19,21);(10,16)(14,23)(15,24)(17,22)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72285487, #72286268, #72286392 |
| maximal normal subgroup of | #72285487, #72286268, #72286392 |
| maximal quotient of | #72285969, #72286030, #72321231, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 104784);