Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #491437 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#493 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #19603 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
602 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 602 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18)(19,27)(20,24)(21,30)(22,32)(23,26)(25,28)(29,31);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19,28,26,22,29,31,32,23,25,27,18,20,30,21,24);(2,7)(6,14)(8,12)(9,15)(17,19,28,26,22,29,31,32,23,25,27,18,20,30,21,24);(17,20,23,22)(18,24,26,32)(19,29,25,30)(21,27,31,28)Polycyclic presentation:
2^9|b^2=eh,c^2=fg,d^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,h^a=hi,d^b=di,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72225185, #72225196, #72225197, … |
| maximal normal subgroup of | #72225185, #72225196, #72225197, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 398655);