Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454029 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2251 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(7,17)(9,10)(12,25)(14,15)(18,27)(19,33)(20,21)(22,24)(26,32)(28,30)(31,34)(35,36);(2,4)(5,6,11,16)(7,18,21,25)(8,13,23,29)(9,15,10,14)(12,17,27,20)(19,32,34,35)(22,30,24,28)(26,31,36,33);(1,2)(3,4)(5,7)(6,12)(8,19)(9,20)(10,17)(11,21)(13,26)(14,18)(15,25)(16,27)(22,31)(23,34)(24,33)(28,35)(29,36)(30,32);(2,4)(5,8)(6,13)(7,19)(9,24)(10,22)(11,23)(12,26)(14,30)(15,28)(16,29)(17,31)(18,32)(20,33)(21,34)(25,35)(27,36)Polycyclic presentation:
2^9|b^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72287194, #72313106, #72313116, … |
| maximal normal subgroup of | #72287194, #72313106, #72313116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361226);