Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #448650 |
| 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 | |
969 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)(5,20)(9,25)(10,22)(13,29)(14,15)(16,26)(17,18)(21,24)(27,31)(30,32);(1,2,5,7,6,10,20,22)(3,12,11,21,15,8,14,24)(4,16,18,9,19,25,17,26)(13,30,29,23,31,32,27,28);(1,3)(2,8)(4,13)(5,14)(6,15)(7,21)(9,23)(10,12)(11,20)(16,32)(17,29)(18,27)(19,31)(22,24)(25,30)(26,28);(1,4)(2,9,10,26)(3,13)(5,17)(6,19)(7,16,22,25)(8,23,12,28)(11,27)(14,29)(15,31)(18,20)(21,32,24,30)Polycyclic presentation:
2^9|b^2=f,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237064, #72237067, #72237084, … |
| maximal normal subgroup of | #72237064, #72237067, #72237084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 355846);