Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #146581 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10735 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1717 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 1717 conjugacy classes of subgroups — too many to draw. Only its 200 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:
(1,2,5,9,6,10,19,27)(3,11,14,28,13,23,17,31)(4,8,16,25,15,24,18,26)(7,20,22,29,21,30,32,12);(1,3)(2,7,10,21)(4,12,15,29)(5,17,19,14)(6,13)(8,23)(9,22)(11,24)(16,30)(18,20)(25,31,26,28)(27,32);(4,15)(8,24)(11,23)(12,29)(16,18)(20,30)(25,26)(28,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,h^b=hi,d^c=di,f^d=fi,g^d=gi,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72303426, #72303432, #72303438 |
| maximal normal subgroup of | #72303426, #72303432, #72303438 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53780);