Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141751 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1957 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #492 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
360 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(17,18,23,26,24,28,22,27)(19,20,29,21,30,32,25,31);(2,6)(3,10)(8,13)(11,16)(17,19,21,28,23,25,32,26,24,30,31,18,22,29,20,27);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(17,20,22,31,24,32,23,21)(18,19,26,29,28,30,27,25)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,c^2=e,d^2=g,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ch,d^b=di,f^b=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225285, #72225287, #72225295, … |
| maximal normal subgroup of | #72225285, #72225287, #72225295, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48968);