Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453972 |
| 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 |
| prime radical | #60369 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2187 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)(5,6,9,15)(7,17,20,12)(8,22,23,28)(10,13,11,16)(14,24,29,25)(18,30,21,27)(19,33,35,36)(26,32,31,34);(2,4)(5,7)(6,12)(8,19)(9,20)(10,18)(11,21)(13,27)(14,26)(15,17)(16,30)(22,36)(23,35)(24,34)(25,32)(28,33)(29,31);(1,2)(3,4)(6,13)(7,18)(8,23)(10,11)(14,22)(15,16)(19,34)(20,21)(26,31)(27,30)(28,29)(32,35);(2,4)(5,8)(6,14)(7,19)(9,23)(10,25)(11,24)(12,26)(13,28)(15,29)(16,22)(17,31)(18,32)(20,35)(21,34)(27,33)(30,36)Polycyclic presentation:
2^9|a^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361174);