Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493477 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20585 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #62862 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #3485 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2301 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)(3,8)(4,9)(5,16,17,7)(6,10)(11,22)(12,21)(13,20,26,18)(14,28,29,19)(15,24)(23,25)(27,31,32,30);(2,7)(3,11)(8,20)(9,19)(10,16)(12,25)(13,26)(18,22)(21,31)(23,30)(24,28)(27,32);(1,3,6,11)(2,8,10,22)(4,12,15,25)(5,13,17,26)(7,18,16,20)(9,23,24,21)(14,27,29,32)(19,31,28,30);(1,4)(2,9)(3,12)(5,14)(6,15)(7,19)(8,21)(10,24)(11,25)(13,27)(16,28)(17,29)(18,30)(20,31)(22,23)(26,32)Polycyclic presentation:
2^9|a^2=g,c^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci,g^b=gh,d^c=dh,f^d=fi,g^e=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72232878, #72240648, #72249652, … |
| maximal normal subgroup of | #72232878, #72240648, #72249652, … |
| maximal quotient of | #72232850, #72233253, #72237441, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400674);