Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,16)(8,20,22,30)(9,23)(10,15)(11,14)(12,18)(13,21,29,25)(17,32,35,26)(19,28)(24,27)(31,34,36,33);(1,2,3,4)(5,7)(6,12)(8,17)(9,24)(10,18)(11,19)(13,26)(14,27)(15,28)(16,23)(20,31)(21,34)(22,35)(25,33)(29,32)(30,36);(5,8)(6,13)(7,17)(9,21)(10,25)(11,22)(12,26)(14,29)(15,30)(16,31)(18,33)(19,35)(20,23)(24,34)(27,32)(28,36);(6,14)(12,27)(13,29)(15,23)(16,28)(20,30)(26,32)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,e^b=ei,f^c=fi,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72250089, #72250582, #72250584, … |
| maximal normal subgroup of | #72250089, #72250582, #72250584, … |
| maximal quotient of | #72240819, #72240821, #72240822, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260650);