Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35149 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2490 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)(5,6,10,15,11,16,9,14)(7,12,19,28,20,29,18,27)(8,21,23,13,25,31,24,32)(17,30,33,35,34,36,22,26);(2,4)(5,7,10,19,11,20,9,18)(6,12,15,28,16,29,14,27)(8,22,23,17,25,33,24,34)(13,30,31,35,32,36,21,26);(5,8)(6,13)(7,17)(9,23)(10,24)(11,25)(12,26)(14,31)(15,21)(16,32)(18,33)(19,22)(20,34)(27,30)(28,36)(29,35);(2,4)(5,9,11,10)(6,14,16,15)(7,18,20,19)(8,23,25,24)(12,27,29,28)(13,31,32,21)(17,33,34,22)(26,30,35,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239541, #72256000, #72256257, … |
| maximal normal subgroup of | #72239541, #72256000, #72256257, … |
| maximal quotient of | #72239574, C2×#364426, #72317024, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271622);