Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2201 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)(5,6,9,14,11,15,10,25)(7,12,18,28,17,27,19,34)(8,20,23,30,22,31,24,13)(16,29,33,35,32,36,21,26);(5,7)(6,12)(8,21,22,33)(9,18)(10,19)(11,17)(13,29,30,36)(14,28)(15,27)(16,24,32,23)(20,35,31,26)(25,34);(5,8)(6,13)(7,16)(9,24)(10,23)(11,22)(12,26)(14,31)(15,30)(17,32)(18,21)(19,33)(20,25)(27,35)(28,36)(29,34);(2,4)(7,17)(12,27)(16,32)(18,19)(21,33)(26,35)(28,34)(29,36)Polycyclic presentation:
2^9|a^2=f,b^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=di,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72258208, #72258209, #72258213, … |
| maximal normal subgroup of | #72258208, #72258209, #72258213, … |
| maximal quotient of | #72258241, #72258255 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302719);