Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394275 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1481 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)(7,15)(9,21)(11,24)(13,28)(16,29)(17,33)(18,22)(19,23)(25,36)(26,30)(31,34)(32,35);(1,2,4,3)(5,6)(7,16,18,11)(8,12)(9,13)(10,14)(15,29,22,24)(17,32,34,25)(19,26)(20,27)(21,28)(23,30)(31,36,33,35);(1,3,4,2)(5,7)(6,11)(8,17)(9,22)(10,18)(12,25)(13,29)(14,16)(15,21)(19,31)(20,34)(23,33)(24,28)(26,35)(27,32)(30,36);(5,8,10,20)(6,12,14,27)(7,17,18,34)(9,23,21,19)(11,25,16,32)(13,30,28,26)(15,31,22,33)(24,35,29,36)Polycyclic presentation:
2^9|b^2=eh,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318058, #72318090, #72318131, … |
| maximal normal subgroup of | #72318058, #72318090, #72318131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312674);