Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #251827 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #47563 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1178 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1178 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,15,16,14)(17,18,21,24)(19,22,26,29)(20,23,27,31)(25,30,32,28);(1,3,6,2)(4,7)(5,10,13,8)(9,12)(11,14)(15,16)(17,19)(18,22)(20,25)(21,26)(23,30)(24,29)(27,32)(28,31);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,20,21,27)(18,23,24,31)(19,25,26,32)(22,28,29,30);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)Polycyclic presentation:
2^9|a^2=hi,b^2=eh,c^2=i,d^2=h,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72284964, #72285102, #72285218, … |
| maximal normal subgroup of | #72284964, #72285102, #72285218, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 159053);