Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#97 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
752 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)(3,6)(4,5)(7,8)(10,16,18,20)(11,21)(13,22)(17,19,24,23);(2,5)(6,8)(9,10,13,19,14,20,21,24,15,18,22,23,12,16,11,17);(10,16,18,20)(11,21)(13,22)(17,19,24,23);(1,3,4,7)(2,6,5,8)(9,11,12,22,15,21,14,13)(10,17,16,23,18,24,20,19)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,d^a=di,g^a=gi,c^b=ch,f^b=fi,d^c=di,g^c=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72213022, C2×#493177, #72216494, … |
| maximal normal subgroup of | #72213022, C2×#493177, #72216494, … |
| maximal quotient of | C2×#493177, #72309908, C4×#12605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400373);