Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2026 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)(7,16)(8,13)(9,24)(10,14)(11,15)(12,18)(17,33)(19,32)(20,27)(21,26)(22,29)(23,31)(25,30)(28,34)(35,36);(5,7,9,18,11,19,10,20)(6,12,14,16,15,27,24,32)(8,21,23,34,22,35,25,17)(13,28,30,26,29,33,31,36);(5,8)(6,13)(7,17,19,34)(9,25)(10,23)(11,22)(12,26,27,36)(14,31)(15,29)(16,28,32,33)(18,35,20,21)(24,30);(8,22)(13,29)(17,34)(21,35)(23,25)(26,36)(28,33)(30,31)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,f^b=fi,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224747, #72317358, #72317367, … |
| maximal normal subgroup of | #72224747, #72317358, #72317367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265673);