Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#390 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2850 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)(8,21)(9,25)(10,14)(11,15)(12,18)(13,23)(17,26)(19,28)(20,27)(22,35)(24,29)(30,33)(31,36)(32,34);(5,7,10,18,11,20,9,19)(6,12,14,27,15,28,25,16)(8,17,24,32,22,31,23,33)(13,26,21,34,29,36,35,30);(2,4)(5,8,9,23,11,22,10,24)(6,13,14,21,15,29,25,35)(7,17,19,33,20,31,18,32)(12,26,27,34,28,36,16,30);(8,22)(13,29)(17,31)(21,35)(23,24)(26,36)(30,34)(32,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72237647, #72237649, #72237669, … |
| maximal normal subgroup of | #72237647, #72237649, #72237669, … |
| maximal quotient of | #72239287, #72239290, #72239292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261310);