Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #448644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15914 |
| 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 | |
956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(5,17)(8,22)(12,24)(13,29)(14,18)(16,19)(23,31)(25,26)(27,30)(28,32);(1,2,5,10,6,7,17,21)(3,12,11,22,14,8,18,24)(4,9,16,25,15,20,19,26)(13,28,29,32,30,23,27,31);(1,3)(2,8)(4,13)(5,18)(6,14)(7,12)(9,23)(10,22)(11,17)(15,30)(16,27)(19,29)(20,28)(21,24)(25,32)(26,31);(1,4)(2,9,7,20)(3,13)(5,19)(6,15)(8,23,12,28)(10,26,21,25)(11,27)(14,30)(16,17)(18,29)(22,31,24,32)Polycyclic presentation:
2^9|b^2=f,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72338466, #72338485, #72338492, … |
| maximal normal subgroup of | #72338466, #72338485, #72338492, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 355840);