Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #452539 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16051 |
| 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 | |
910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,11,6,12,7,22)(3,9,14,29,5,20,21,25)(4,15,19,27,18,10,23,28)(13,30,31,32,17,24,16,26);(1,3)(2,9)(4,16,18,31)(5,6)(7,14)(8,21)(10,26,15,32)(11,25)(12,20)(13,19,17,23)(22,29)(24,27,30,28);(1,4,6,18)(2,10)(3,13,5,17)(7,23,8,19)(9,24)(11,28)(12,15)(14,31,21,16)(20,30)(22,27)(25,32)(26,29);(1,5,6,3)(2,9)(4,17,18,13)(7,14,8,21)(10,24)(11,29)(12,20)(15,30)(16,23,31,19)(22,25)(26,28)(27,32)Polycyclic presentation:
2^9|a^2=egh,b^2=f,c^2=ef,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72319099, #72319370, #72319453, … |
| maximal normal subgroup of | #72319099, #72319370, #72319453, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 359735);