Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152710 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13086 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | #23915 | C2 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
676 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 676 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)(3,13,17,28,16,30,18,32)(4,19,5,9)(6,12)(7,11)(8,15,25,21,27,31,26,14)(10,22,24,20)(23,29);(1,3,4,14)(2,8,24,32)(5,21,7,16)(6,17,22,15)(9,13,12,25)(10,28,11,27)(18,20,31,23)(19,30,29,26);(1,4)(2,9)(3,14)(5,7)(6,20)(8,13)(10,29)(11,19)(12,24)(15,18)(16,21)(17,31)(22,23)(25,32)(26,28)(27,30)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,d^b=dg,e^b=ei,f^b=fg,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 | #72213534, #72219528, #72238420, … |
| maximal normal subgroup of | #72213534, #72219528, #72238420, … |
| maximal quotient of | #72250820, C2×#152710 |
| automorphism group of | #1412, #1413 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59906);