Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
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×#385 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,14,9,22)(7,16,18,12,19,27,23,26)(8,13,20,30,21,28,24,29)(17,32,35,36,34,31,33,25);(1,2,3,4)(5,7)(6,12)(8,17)(9,23)(10,18)(11,19)(13,25)(14,26)(15,27)(16,22)(20,35)(21,34)(24,33)(28,36)(29,31)(30,32);(5,8,10,20,11,21,9,24)(6,13,15,30,14,28,22,29)(7,17,18,35,19,34,23,33)(12,25,27,32,26,36,16,31);(6,14)(12,26)(13,28)(15,22)(16,27)(25,36)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,e^b=ei,f^b=fi,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72252182, #72252190, #72252194, … |
| maximal normal subgroup of | #72252182, #72252190, #72252194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261110);