Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93193 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1316 |
| commutator | C22×Q8 | C2×C8 |
| Frattini, cosocle | #1446 | C22 |
| prime radical | #3486 | C4 |
| LCS | G ▷ C22×Q8 ▷ C22×C4 ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ #1446 ◁ G | |
| LECS | G ▷ #1446 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C1 | |
318 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 30 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) = 64 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,8)(3,9,10,16,11,6,12,14)(5,7,13,15)(18,22,23,31,25,27,24,32)(19,26,28,29);(1,3)(2,6)(4,12)(5,11)(7,9)(8,14)(10,13)(15,16)(17,18)(19,27)(20,23)(21,25)(22,28)(24,30)(26,31)(29,32)Polycyclic presentation:
2^9|a^2=d,c^2=i,d^2=f,e^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=ci,d^b=de,e^b=ei,f^b=fhi,g^b=gi,d^c=dgi,e^c=ei,e^d=eh44 conjugacy classes — view the full character table.
| maximal subgroup of | #72267806, #72267850 |
| maximal normal subgroup of | #72267806, #72267850 |
| maximal quotient of | #72267812 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389);