Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| 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 | |
4434 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)(5,6,10,14,11,16,9,15)(7,17,20,12,19,29,21,28)(8,22,25,13,26,31,24,32)(18,33,35,30,34,27,23,36);(2,4)(5,7)(6,12)(8,23)(9,21)(10,20)(11,19)(13,30)(14,29)(15,17)(16,28)(18,25)(22,33)(24,34)(26,35)(27,31)(32,36);(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,31)(15,22)(16,32)(17,30)(19,34)(20,35)(21,23)(28,33)(29,36);(7,19)(12,28)(17,29)(18,34)(20,21)(23,35)(27,33)(30,36)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224751, #72237504, #72237513, … |
| maximal normal subgroup of | #72224751, #72237504, #72237513, … |
| maximal quotient of | #72257655, #72257672, #72257747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264742);