Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417579 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60454 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1743 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)(7,11)(8,12)(9,21)(10,14)(13,18)(15,24)(16,28)(17,26)(19,32)(20,30)(22,31)(23,27)(25,33)(29,34)(35,36);(1,2)(3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,28)(13,21)(14,26)(15,27)(19,34)(20,33)(22,35)(23,24)(25,30)(29,32)(31,36);(5,8,10,20)(6,12,14,30)(7,16,17,33)(9,22,24,19)(11,25,26,28)(13,29,27,36)(15,32,21,31)(18,35,23,34);(1,3,4,2)(5,7)(6,13)(8,16)(9,23)(10,17)(11,21)(12,29)(14,27)(15,26)(18,24)(19,35)(20,33)(22,34)(25,31)(28,32)(30,36)Polycyclic presentation:
2^9|c^2=i,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325522);