Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #180873 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#97 |
| prime radical | C22×D4 | M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,16,11,18,28,34,12,17,27,35,9,14,26,31)(7,19,22,36,23,13,29,30,20,8,15,25,21,24,32,33);(2,4)(5,7)(6,13,17,24)(8,14,19,18)(9,21)(10,15)(11,23)(12,20)(16,33,35,30)(22,27)(25,34,36,31)(26,29)(28,32);(6,14,17,18)(7,20)(8,24,19,13)(10,27)(16,34,35,31)(21,23)(25,30,36,33)(26,28);(5,8,10,25,11,24,28,33,12,19,27,36,9,13,26,30)(6,15,16,21,18,32,34,7,17,22,35,23,14,29,31,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=i,f^2=h,h^2=i,b^a=be,c^a=ch,g^a=gi,c^b=ci,d^b=dh,f^b=fi,d^c=dh,f^c=fi,g^d=gi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72313670, #72313701, #72313712, … |
| maximal normal subgroup of | #72313670, #72313701, #72313712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400421);