Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×C4⋊C8 |
| prime radical | C23 | #431 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
308 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6,8,11)(7,13,14,15);(1,2)(3,7)(4,11)(5,8)(6,12)(9,15)(10,14)(13,16)(17,18,19,23,20,24,25,30,21,22,26,28,27,29,31,32);(1,3,4,9,5,10,12,16)(2,7,11,15,8,14,6,13)(18,22)(23,28)(24,29)(30,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=g,f^2=h,h^2=i,b^a=bd,c^b=ci,d^b=dg,e^b=eg176 conjugacy classes — view the full character table.
| maximal subgroup of | #72225412, #72225507, #72225520, … |
| maximal normal subgroup of | #72225412, #72225507, #72225520, … |
| maximal quotient of | #72225407 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36272);