Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478455 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17149 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23445 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12,14,7)(4,8,17,25)(5,11)(6,10)(9,15,28,20)(13,23)(16,22)(18,26,19,30)(21,29)(24,32)(27,31);(1,3)(2,7)(4,16,17,13)(5,15)(6,14)(8,23,25,22)(9,11)(10,12)(18,32,19,31)(20,21)(24,26,27,30)(28,29);(1,4)(2,8,10,25)(3,13)(5,19)(6,17)(7,22,12,23)(9,27,28,24)(11,30,29,26)(14,16)(15,32)(18,21)(20,31);(1,3)(2,9)(4,13)(5,20)(6,14)(7,11)(8,24)(10,28)(12,29)(15,21)(16,17)(18,32)(19,31)(22,26)(23,30)(25,27)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,g^b=gi,d^c=de,f^c=fi,h^c=hi,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72285906, #72286066, #72286126, … |
| maximal normal subgroup of | #72285906, #72286066, #72286126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385651);