Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480656 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,10,5,9,15,23)(3,8,12,22,11,21,24,32)(6,7,14,19,16,18,27,29)(13,17,26,28,25,20,30,31);(2,7)(3,11)(8,20)(9,18)(12,24)(13,25)(17,21)(22,32)(26,30)(28,31);(1,3,6,13)(2,8,7,17)(4,12,14,26)(5,11,16,25)(9,21,18,20)(10,22,19,28)(15,24,27,30)(23,32,29,31);(2,9)(7,18)(8,21)(10,23)(17,20)(19,29)(22,32)(28,31)Polycyclic presentation:
2^9|a^2=f,c^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,f^b=fg,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72278098, #72278134, #72278179, … |
| maximal normal subgroup of | #72278098, #72278134, #72278179, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387854);