Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461095 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16379 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
903 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,6,12,5,10)(3,13,14,24,16,22,17,8)(4,18,19,26,20,9,21,25)(15,27,28,31,30,32,29,23);(1,3,6,16)(2,8)(4,15,20,30)(5,14,7,17)(9,23)(10,13)(11,22)(12,24)(18,31)(19,29,21,28)(25,32)(26,27);(3,14)(4,19)(5,7)(8,22)(9,25)(10,11)(13,24)(16,17)(18,26)(20,21)(23,31)(28,29);(1,4)(2,9)(3,15)(5,21)(6,20)(7,19)(8,23)(10,26)(11,25)(12,18)(13,27)(14,28)(16,30)(17,29)(22,32)(24,31)Polycyclic presentation:
2^9|a^2=fg,b^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237068, #72237069, #72237086, … |
| maximal normal subgroup of | #72237068, #72237069, #72237086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368291);