Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #148011 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11043 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
918 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 918 conjugacy classes of subgroups — too many to draw. Only its 132 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)(3,7)(4,14,16,25)(5,18,20,9)(6,10)(8,17)(11,26,15,31)(12,28,30,22)(13,23)(19,24)(21,27)(29,32);(1,3)(2,7)(4,15,16,11)(5,12)(6,13)(8,21,24,32)(9,22)(10,23)(14,26,25,31)(17,29,19,27)(18,28)(20,30);(1,4)(2,8)(3,11)(5,19)(6,16)(7,21)(9,25)(10,24)(12,29)(13,15)(14,18)(17,20)(22,31)(23,32)(26,28)(27,30)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,f^b=fi,d^c=di,e^c=ei,g^c=ghi,g^e=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72238714, #72238719, #72238722, … |
| maximal normal subgroup of | #72238714, #72238719, #72238722, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55207);