Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2722 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)(7,12)(8,20)(9,14)(10,24)(11,15)(13,22)(16,31)(17,27)(18,33)(19,28)(21,35)(23,29)(25,30)(26,32)(34,36);(5,7)(6,12)(8,21,23,16)(9,17)(10,18)(11,19)(13,26,30,36)(14,27)(15,28)(20,31,29,35)(22,34,25,32)(24,33);(5,8)(6,13)(7,16)(9,22)(10,25)(11,23)(12,26)(14,29)(15,30)(17,32)(18,34)(19,21)(20,24)(27,35)(28,36)(31,33);(5,9,11,10)(6,14,15,24)(7,17,19,18)(8,22,23,25)(12,27,28,33)(13,29,30,20)(16,32,21,34)(26,35,36,31)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72316334, #72316351, #72316355, … |
| maximal normal subgroup of | #72316334, #72316351, #72316355, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264969);