Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #10336 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
736 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 736 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,9,11,6)(4,12,14,7)(5,8)(10,16)(13,15)(17,18)(19,25,26,22)(20,23,28,30)(21,24)(27,31)(29,32);(1,3,5,11)(2,6,8,9)(4,13,14,10)(7,15,12,16)(19,26)(29,31);(1,4)(2,7)(3,10,11,13)(5,14)(6,15,9,16)(8,12)(17,19)(18,22,24,25)(20,27)(21,26)(23,29,30,31)(28,32);(6,9)(15,16)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=efi,d^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,d^b=de,f^b=fh,e^c=ei,f^c=fi,g^c=gh,g^d=gi38 conjugacy classes — view the full character table.
| Fitting subgroup of | #72284969, #72285016 |
| maximal subgroup of | #72283648, #72283693, #72283821, … |
| maximal normal subgroup of | #72283648, #72283693, #72283821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253471);