Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213760 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #35463 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,15,18,11)(8,19)(9,13,22,28)(10,14)(12,21)(16,25,30,23)(17,24)(20,26,33,32)(27,31)(29,35)(34,36);(1,2)(3,4)(5,7,10,18)(6,11,14,15)(8,16,21,30)(9,17,22,31)(12,25,19,23)(13,27,28,24)(20,29,33,36)(26,34,32,35);(5,8)(6,12,14,19)(7,16)(9,20)(10,21)(11,23,15,25)(13,26,28,32)(17,29)(18,30)(22,33)(24,34,27,35)(31,36);(1,3,4,2)(5,9)(6,13,14,28)(7,17)(8,20)(10,22)(11,24,15,27)(12,26,19,32)(16,29)(18,31)(21,33)(23,34,25,35)(30,36)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283486, #72283557, #72283795, … |
| maximal normal subgroup of | #72283486, #72283557, #72283795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293214);