Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152714 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13086 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | #23913 | C2 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
679 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 679 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)(3,7,13,20,14,21,28,23)(4,15,16,8)(5,9)(6,10)(11,18)(12,19,25,31,26,29,27,32)(17,22,30,24);(1,3,6,14)(2,7,11,20)(4,12,16,26)(5,13)(8,19,22,32)(9,23,10,21)(15,29,24,31)(17,27)(18,28)(25,30);(1,4)(2,8)(3,12)(5,17)(6,16)(7,19)(9,24)(10,15)(11,22)(13,27)(14,26)(18,30)(20,32)(21,29)(23,31)(25,28)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,d^b=dgi,e^b=ei,f^b=fgi,g^b=gh,h^b=hi,d^c=di,f^c=fi,f^d=fhi,g^d=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72213519, #72216601, #72238420, … |
| maximal normal subgroup of | #72213519, #72216601, #72238420, … |
| maximal quotient of | #72250648, C2×#152716, #72286271, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59912);