Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #391 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1441 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,12,18,27)(8,21)(9,16)(10,15)(11,14)(13,23)(17,32,34,36)(19,28,20,26)(22,31)(24,30)(25,35,29,33);(5,7,10,20,11,18,9,19)(6,12,16,28,14,27,15,26)(8,17,24,35,22,34,23,33)(13,29,31,32,30,25,21,36);(1,2,3,4)(5,8,9,23,11,22,10,24)(6,13,15,21,14,30,16,31)(7,17,19,33,18,34,20,35)(12,25,26,32,27,29,28,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=di,g^b=gi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239210, #72239644, #72239645, … |
| maximal normal subgroup of | #72239210, #72239644, #72239645, … |
| maximal quotient of | #72239214, #72249850, C2×#123905, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31101);