Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11769 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1505 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C2 | #8218 |
| prime radical | #3054 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
438 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 438 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(1,2,5,11,7,13,6,12)(3,14,16,28,18,30,17,23)(4,9,20,26,22,27,21,10)(8,19,24,15,25,31,29,32);(1,3,6,16,7,18,5,17)(2,8,12,24,13,25,11,29)(4,19,20,32,22,31,21,15)(9,14,26,23,27,30,10,28);(1,4,5,20,7,22,6,21)(2,9,11,26,13,27,12,10)(3,15,16,31,18,32,17,19)(8,23,24,14,25,28,29,30)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,g^b=gi,d^c=di,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213108, C2×#128835, #72217918 |
| maximal normal subgroup of | C2×#128835, #72217918 |
| maximal quotient of | #72213047, #72213051, #72213052, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36031);