Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93213 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1343 |
| commutator | C2×#95 | C2×C4 |
| Frattini, cosocle | #1446 | C22 |
| prime radical | C24 | #99 |
| LCS | G ▷ C2×#95 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ #1446 ◁ G | |
| LECS | G ▷ #1446 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C1 | |
269 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 269 conjugacy classes of subgroups — too many to draw. Only its 28 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)(3,9)(4,12)(5,8)(6,10,14,16)(7,13)(11,15)(17,18,21,25,23,19,20,28)(22,29,30,24,32,26,31,27);(1,3)(2,6)(4,10)(5,11)(7,15)(8,14)(9,12)(13,16)(17,19,23,18)(20,27,21,24)(22,26,32,29)(25,30,28,31)Polycyclic presentation:
2^9|a^2=d,b^2=h,c^2=f,d^2=h,e^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,g^a=gh,c^b=cf,d^b=defh,e^b=eh,g^b=gi,d^c=dgh,e^c=ei,e^d=eh38 conjugacy classes — view the full character table.
| maximal subgroup of | #72267850 |
| maximal normal subgroup of | #72267850 |
| maximal quotient of | #72267854 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 412);