Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C42 |
| commutator | C4 | C2×C82 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C2×#97 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
362 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 512 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(17,18,21,27,22,24,29,31,23,28,30,32,20,26,19,25);(2,5)(6,8)(9,10,11,13,12,14,15,16)(18,24,28,26)(19,29)(21,30)(25,32,31,27);(1,3,4,7)(2,6,5,8)(17,19,20,30,23,29,22,21)(18,25,26,32,28,31,24,27)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=d,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,f^a=fi,c^b=ci,e^b=ei176 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#1313, #72223574, C2×#128208, … |
| maximal normal subgroup of | C8×#1313, #72223574, C2×#128208, … |
| maximal quotient of | C2×#128208, #72223648 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35404);