Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1216 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)(7,17,21,28,19,27,20,12)(8,13,24,31,25,32,23,30)(18,29,22,35,33,36,34,26);(5,7,11,19)(6,12)(8,22)(9,20,10,21)(13,29,32,36)(14,27)(15,17)(16,28)(18,23)(24,33)(25,34)(26,31,35,30);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,30)(15,31)(16,32)(17,29)(19,33)(20,34)(21,22)(27,36)(28,35);(6,14,16,15)(7,19)(12,17,28,27)(13,30,32,31)(18,33)(20,21)(22,34)(26,29,35,36)Polycyclic presentation:
2^9|a^2=h,b^2=eg,d^2=eh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#378811, #72216224, #72216230, … |
| maximal normal subgroup of | C2×#378811, #72216224, #72216230, … |
| maximal quotient of | #72216007, #72216008, #72216010, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286007);