Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2698 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)(5,6,10,16,11,15,9,14)(7,12,20,29,18,27,19,28)(8,21,25,13,23,31,24,30)(17,32,34,26,22,35,33,36);(5,7)(6,12)(8,22,23,17)(9,19)(10,20)(11,18)(13,26,30,36)(14,28)(15,27)(16,29)(21,32,31,35)(24,34,25,33);(2,4)(5,8)(6,13)(7,17)(9,25)(10,24)(11,23)(12,26)(14,31)(15,30)(16,21)(18,22)(19,34)(20,33)(27,36)(28,35)(29,32);(7,18)(12,27)(17,22)(19,20)(26,36)(28,29)(32,35)(33,34)Polycyclic presentation:
2^9|a^2=h,b^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=di,g^b=gi,e^c=ei,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237172, #72237173, #72237178, … |
| maximal normal subgroup of | #72237172, #72237173, #72237178, … |
| maximal quotient of | #72239010, #72239017, C2×#357771, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264967);