Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | #34547 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6,7,8)(9,10,12,16,13,14,19,21)(11,15,17,22,18,20,23,24);(1,2)(3,6)(4,8)(5,7)(10,14)(15,20);(1,3,5,4)(2,6,7,8);(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=i,e^2=h,f^2=i,b^a=be,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,f^e=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388729);