Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #95 |
| commutator | C2×C4 | C4×C16 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1374 |
| prime radical | C23 | C4⋊C16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2)(3,10,12,16)(4,8)(5,14)(6,9)(7,11,15,13)(17,18,20,26,21,28,31,41,22,25,32,38,34,39,44,47,23,27,33,40,35,42,45,48,19,24,29,36,30,37,43,46);(1,3,5,13,6,12,4,11)(2,7,8,10,9,15,14,16)(18,24,27,25)(26,36,40,38)(28,37,42,39)(41,46,48,47)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=f,g^2=h,h^2=i,b^a=bc,c^a=ci,e^a=efi,d^b=df128 conjugacy classes — view the full character table.
| maximal subgroup of | #72210594, C2×#94006 |
| maximal normal subgroup of | #72210594, C2×#94006 |
| maximal quotient of | C2×#94006 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1202);