Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479840 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17273 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
813 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 813 conjugacy classes of subgroups — too many to draw. Only its 152 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:
(1,2,6,12)(3,13)(4,16,17,31)(5,10)(7,22,19,11)(8,15)(9,24,25,18)(14,23)(20,27)(21,32)(26,28)(29,30);(1,3,6,15)(2,8)(4,10)(5,16,20,31)(7,14,19,30)(9,21,25,28)(11,23)(12,13)(17,27)(18,26)(22,29)(24,32);(1,4,7,18,6,17,19,24)(2,9,11,16,12,25,22,31)(3,10,14,26,15,27,30,32)(5,8,21,23,20,13,28,29);(1,5,19,28,6,20,7,21)(2,10,11,26,12,27,22,32)(3,9,14,16,15,25,30,31)(4,13,24,23,17,8,18,29)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,h^c=hi,g^d=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72251676, #72251688, #72251710, … |
| maximal normal subgroup of | #72251676, #72251688, #72251710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387036);