Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500599 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13341 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
665 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 665 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,16)(8,22)(9,17)(10,27)(11,15)(12,18)(13,20)(14,19)(21,30)(23,33)(24,31)(25,36)(26,29)(28,32)(34,35);(1,2,3,4)(5,7,11,10,12,21,9,20)(6,13,17,30,18,27,15,16)(8,23,24,35,26,28,25,19)(14,31,32,22,34,36,33,29);(2,4)(5,8)(6,14)(7,19)(9,24)(10,28)(11,25)(12,26)(13,29)(15,32)(16,31)(17,33)(18,34)(20,23)(21,35)(22,27)(30,36);(5,7,9,20,12,21,11,10)(6,13,15,16,18,27,17,30)(8,19,24,23,26,35,25,28)(14,29,32,31,34,22,33,36)Polycyclic presentation:
2^9|b^2=g,d^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72265538, #72265587, #72265615, … |
| maximal normal subgroup of | #72265538, #72265587, #72265615, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 413497);