Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499266 |
| 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 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
714 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 714 conjugacy classes of subgroups — too many to draw. Only its 146 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:
(5,6,8,15,25,14,27,16,11,17,24,13,10,18,9,26)(7,19,21,30,28,31,36,32,20,34,35,33,23,29,22,12);(1,2)(3,4)(5,7,11,20)(6,12,17,32)(8,22,24,36)(9,21,27,35)(10,28,25,23)(13,31,15,29)(14,33,18,30)(16,34,26,19);(2,4)(6,13)(8,9)(10,25)(12,29)(14,16)(15,17)(18,26)(19,33)(21,22)(23,28)(24,27)(30,34)(31,32)(35,36);(6,14,17,18)(7,20)(8,24)(9,27)(12,30,32,33)(13,16,15,26)(19,29,34,31)(23,28)Polycyclic presentation:
2^9|a^2=e,b^2=i,d^2=ef,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^b=eh,f^b=fhi,h^b=hi,e^c=eh,f^c=fhi,h^c=hi,e^d=ei,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72224928, #72252778, #72265562, … |
| maximal normal subgroup of | #72224928, #72252778, #72265562, … |
| maximal quotient of | #72224925 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406856);