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 | C4×#120 |
| prime radical | #18516 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1095 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,13,17,14,11,18,16)(12,15,20,23,21,19,24,22);(2,6)(3,4)(7,8)(9,11)(10,14)(12,19)(13,16)(15,21)(17,18)(20,22)(23,24);(9,10)(11,14)(12,15)(13,17)(16,18)(19,21)(20,23)(22,24);(1,3,5,4)(2,7,8,6)(9,12,13,20,14,21,18,24)(10,15,17,23,11,19,16,22)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,e^b=eh,f^b=fi,e^c=ei,f^c=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72285370, #72286208, #72286221, … |
| maximal normal subgroup of | #72285370, #72286208, #72286221, … |
| maximal quotient of | #72285475, #72285489, #72287298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 117552);