Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(6,8)(9,10,14,12,15,18,13,17)(11,19,22,21,23,20,24,16);(2,4)(9,11)(10,16)(12,20)(13,22)(14,24)(15,23)(17,19)(18,21);(5,6,7,8)(9,10,13,17,15,18,14,12)(11,20,24,21,23,19,22,16);(9,12,14,18,15,17,13,10)(11,21,22,20,23,16,24,19);(9,13,15,14)(10,17,18,12)(11,22,23,24)(16,19,21,20)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=h,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372815);