Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #427 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C2 | #12013 |
| prime radical | C23 | #364 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ #346 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
144 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,14,8,13,20,31,9,15,18,28,5,25,21,29,10,16,26,32,24,12,19,30,7,27,23,11,4,22,3,17);(1,3,5,18,7,19,8,6,10,21,4,23,9,20,24,26)(2,11,13,17,15,14,25,31,16,28,12,29,27,32,22,30);(1,4,7,24,10,5,9,8)(2,12,15,22,16,13,27,25)(3,18,19,6,21,23,20,26)(11,17,14,31,28,29,32,30)Polycyclic presentation:
2^9|a^2=e,b^2=d,c^2=f,d^2=f,e^2=g,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,c^b=ci,e^b=ei,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72214100, #72214147, #72214160, … |
| maximal normal subgroup of | #72214100, #72214147, #72214160, … |
| maximal quotient of | #72216243, #72216247, C3×#151045, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58241);