Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7225 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | H(3,ℤ/4) |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C4×M32 | C22 |
| socle | C2 | #7146 |
| prime radical | C23 | #364 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,10,5,13,20,28,6,14,21,31,18,8,19,24,7,12,22,30,23,9,25,26,16,11,32,29,3,15,17,27);(2,8,11,13,12,15,14,9)(10,27,29,26,30,24,31,28)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,e^2=f,g^2=h,h^2=i,b^a=bc,c^a=ch,e^a=efhi,f^a=fi,d^b=df,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#94008, #72210674, #72210675 |
| maximal normal subgroup of | C2×#94008, #72210674, #72210675 |
| maximal quotient of | #72210598, C2×#94008 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1204);