Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420461 |
| 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 | #23443 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,16,10,17,11,25)(7,13,9,15,19,28,20,32)(8,21,26,31,23,35,24,14)(18,30,22,29,33,36,34,27);(1,2)(3,4)(5,7)(6,13)(8,18)(9,11)(10,19)(12,20)(14,29)(15,25)(16,32)(17,28)(21,30)(22,24)(23,33)(26,34)(27,31)(35,36);(2,4)(5,8,10,23)(6,14)(7,18,19,33)(9,22,20,34)(11,24,12,26)(13,27)(15,30)(16,21)(17,31)(25,35)(28,29)(32,36);(5,9)(6,15,17,32)(7,12)(8,22)(10,20)(11,19)(13,25,28,16)(14,30,31,36)(18,26)(21,27,35,29)(23,34)(24,33)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72316311, #72316345, #72316355 |
| maximal normal subgroup of | #72316311, #72316345, #72316355 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327682);