Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #2403 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
643 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 643 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)(4,12,13,7)(5,8)(10,15)(14,16)(17,18)(19,27,29,23)(20,30)(21,28,32,25)(22,26)(24,31);(1,3)(2,6)(4,10)(5,11)(7,15)(8,9)(12,14)(13,16)(17,19,22,29)(18,23,26,27)(20,25,31,28)(21,24,32,30);(1,4)(2,7)(3,10,11,16)(5,13)(6,14,9,15)(8,12)(17,20)(18,24,26,30)(19,28,29,25)(21,23)(22,31)(27,32);(6,9)(14,15)(17,21)(18,25)(19,24)(20,23)(22,32)(26,28)(27,31)(29,30)Polycyclic presentation:
2^9|a^2=fghi,b^2=i,c^2=ef,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.
| maximal subgroup of | #72283465, #72283491, #72283568, … |
| maximal normal subgroup of | #72283465, #72283491, #72283568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253453);