Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152678 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13082 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | #23908 | C2 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
623 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 623 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(1,2,5,9,7,11,4,18)(3,13,17,31,16,8,23,26)(6,10,19,28,22,12,21,24)(14,30,15,27,20,25,32,29);(1,3,5,20)(2,8,24,26,11,13,28,31)(4,14,7,16)(6,15,21,17)(9,25,10,27,18,30,12,29)(19,23,22,32);(1,4)(2,9)(3,14)(5,7)(6,19)(8,25)(10,24)(11,18)(12,28)(13,30)(15,23)(16,20)(17,32)(21,22)(26,27)(29,31)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dgi,e^b=ei,f^b=fgi,g^b=gh,d^c=di,f^c=fi,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72279292, #72279336, #72279891, … |
| maximal normal subgroup of | #72279292, #72279336, #72279891, … |
| maximal quotient of | C2×#152680 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59876);