Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1853 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,15,16,27)(8,19)(9,22)(10,13)(11,17,24,29)(12,21)(14,23)(18,32,30,28)(20,33)(25,36,35,31)(26,34);(5,7,8,17,9,18,21,31,10,16,20,29,23,30,34,36)(6,11,12,15,14,25,19,28,13,24,26,27,22,35,33,32);(1,2,3,4)(7,16)(8,20)(12,26)(15,27)(18,30)(19,33)(21,34)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,f^b=fi,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#149193, #72214486, #72214501, … |
| maximal normal subgroup of | C2×#149193, #72214486, #72214501, … |
| maximal quotient of | #72214315, C2×#149193, #72215302 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56389);