Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
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 | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4398 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)(7,17)(8,13)(9,14)(10,15)(11,16)(12,20)(18,33)(19,27)(21,28)(22,36)(23,30)(24,32)(25,31)(26,34)(29,35);(5,7)(6,12)(8,22)(9,19)(10,20)(11,21)(13,29)(14,17)(15,28)(16,27)(18,23)(24,34)(25,35)(26,31)(30,36)(32,33);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,30)(15,31)(16,32)(17,29)(19,34)(20,22)(21,35)(27,36)(28,33);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,23,24,25)(12,17,27,28)(13,30,32,31)(18,34,35,22)(26,29,36,33)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72226729, #72237177, #72237184, … |
| maximal normal subgroup of | #72226729, #72237177, #72237184, … |
| maximal quotient of | #72226722, #72239067, #72239837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264789);