Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93313 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1281 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C4×M32 | C22 |
| socle | C2 | #7196 |
| prime radical | C23 | #347 |
| LCS | G ▷ C2×C8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C4×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C4×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,10,5,13,22,30,6,14,19,28,16,9,20,26,7,12,21,29,23,8,24,25,17,11,31,32,3,15,18,27);(2,8,11,9,12,13,14,15)(3,16)(4,19,21,31)(5,23)(10,26,28,25,29,27,32,30)(18,24,20,22)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=hi,d^2=g,e^2=f,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=eh,f^a=fi,c^b=ci,d^b=dfh,g^b=gi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286095, #72286118, #72286160, … |
| maximal normal subgroup of | #72286095, #72286118, #72286160, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1317);