Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419895 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1203 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)(5,6,9,14,11,16,10,15)(7,17,20,12,21,29,19,28)(8,22,25,31,26,30,24,13)(18,27,33,32,23,35,34,36);(5,7,9,20,11,21,10,19)(6,12,15,17,16,28,14,29)(8,23,24,33,26,18,25,34)(13,27,22,32,31,35,30,36);(5,8)(6,13,16,31)(7,18,21,23)(9,24)(10,25)(11,26)(12,27)(14,30,15,22)(17,32)(19,33,20,34)(28,35)(29,36);(2,4)(6,14)(7,19)(9,10)(12,28)(13,30)(15,16)(18,33)(20,21)(22,31)(23,34)(24,25)(27,35)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285470, #72285507, #72285600 |
| maximal normal subgroup of | #72285470, #72285507, #72285600 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341263);