Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245282 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1503 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 1503 conjugacy classes of subgroups — too many to draw. Only its 148 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)(3,6)(4,7)(5,8)(9,14)(10,13)(11,15)(12,16)(20,27)(22,29)(23,31)(30,32);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(17,18)(19,22)(20,23)(21,24)(25,30)(26,29)(27,31)(28,32);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,19)(18,22)(20,28)(21,26)(23,32)(24,29)(25,27)(30,31);(4,11)(7,15)(9,16)(12,14)(17,20,21,27)(18,23,24,31)(19,25,26,28)(22,30,29,32)Polycyclic presentation:
2^9|b^2=hi,d^2=h,b^a=be,c^a=cf,d^a=deh,g^a=ghi,c^b=cg,f^b=fi,d^c=di,e^c=eh,f^d=fh44 conjugacy classes — view the full character table.
| maximal subgroup of | #72284232, #72284327, #72284452, … |
| maximal normal subgroup of | #72284232, #72284327, #72284452, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152487);