Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #449385 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15941 |
| 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 | |
859 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)(3,15)(4,10)(5,20)(6,13)(7,12)(8,14)(9,17)(11,24)(16,31)(18,25)(19,32)(21,28)(22,30)(23,29)(26,27);(1,3,6,17,8,10,7,18)(2,9,12,15,14,25,13,4)(5,21,24,16,23,32,22,27)(11,28,20,26,30,19,29,31);(1,4,6,9,8,15,7,25)(2,10,13,17,14,3,12,18)(5,19,22,26,23,28,24,31)(11,27,20,21,30,16,29,32);(1,5,8,23)(2,11)(3,16)(4,19,15,28)(6,22,7,24)(9,26,25,31)(10,27)(12,29)(13,20)(14,30)(17,21)(18,32)Polycyclic presentation:
2^9|b^2=ef,c^2=g,d^2=eg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251690, #72251692, #72251699, … |
| maximal normal subgroup of | #72251690, #72251692, #72251699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356581);