Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(9,10)(11,15)(12,14)(13,16)(17,18,22,25,23,26,21,19)(20,28,30,24,32,27,31,29);(5,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,21,26,23,25,22,18)(20,29,31,27,32,24,30,28);(1,2,3,4)(6,8)(17,20)(18,24)(19,27)(21,30)(22,31)(23,32)(25,28)(26,29);(10,14)(13,15)(17,21,23,22)(18,19,26,25)(20,30,32,31)(24,27,29,28)Polycyclic presentation:
2^9|a^2=h,b^2=egi,c^2=fh,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fi,g^c=gi,h^c=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72264913, #72264920, #72264952 |
| maximal normal subgroup of | #72264913, #72264920, #72264952 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282345);