Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346276 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2840 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
689 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 689 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,15,16,14)(22,30)(31,32);(1,3)(2,6,8,9)(4,10)(5,11)(7,15,12,14)(13,16)(17,18)(19,22)(20,27,29,23)(21,24)(25,31,28,32)(26,30);(1,4)(2,7,8,12)(3,10)(5,13)(6,14,9,15)(11,16)(17,19)(18,22)(20,28,29,25)(21,26)(23,32,27,31)(24,30);(10,16)(14,15)(17,20)(18,23)(19,25)(21,29)(22,31)(24,27)(26,28)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=def,e^b=eh,f^b=fhi,d^c=de,e^c=ei,f^c=fh,g^d=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72283654, #72283815, #72283854, … |
| maximal normal subgroup of | #72283654, #72283815, #72283854, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253473);