Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511331 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32468 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #525 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
400 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 112 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 — 4 generators on 32 points:
(1,2)(3,13,17,10)(4,7)(5,9)(6,12)(8,11)(14,30,31,27)(15,23,24,16)(18,20)(19,25)(21,29,32,26)(22,28);(2,9)(3,14,17,31)(4,11)(5,7)(6,20)(10,26,13,29)(15,30,24,27)(16,32,23,21)(18,25)(19,22);(1,3,5,16,8,17,7,23)(2,10,9,24,11,13,4,15)(6,21,19,31,22,32,20,14)(12,26,25,30,28,29,18,27);(1,4,8,9)(2,7,11,5)(3,15,17,24)(6,18,22,25)(10,23,13,16)(12,20,28,19)(14,29,31,26)(21,27,32,30)Polycyclic presentation:
2^9|a^2=efh,b^2=efh,c^2=ef,d^2=i,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,g^c=gh,h^c=hi,e^d=ei,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72237280, C2×#511332, #72320520, … |
| maximal normal subgroup of | #72237280, C2×#511332, #72320520, … |
| maximal quotient of | C2×#511332, #72250945, #72257143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418528);