Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | C23 | D4⋊C8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
256 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 256 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6)(3,9)(4,11)(7,16)(8,15)(10,13)(12,14)(17,18,20,26,21,27,30,25,22,24,31,32,23,19,28,29);(1,2,5,8)(3,10,12,16)(4,13,11,7)(6,14,15,9)(17,19,22,27)(18,23,24,21)(20,25,31,29)(26,30,32,28);(18,24)(19,27)(25,29)(26,32)Polycyclic presentation:
2^9|a^2=e,b^2=h,d^2=fhi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,c^b=ci,d^b=dfi,e^b=ei,f^b=fh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72210451 |
| maximal normal subgroup of | #72210451 |
| maximal quotient of | #72210437 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29701);