Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #375190 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14152 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60454 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1540 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)(3,11,15,7)(4,16,17,8)(5,19)(6,9)(10,20)(12,14,23,21)(13,22)(18,24,30,29)(25,27)(26,28)(31,32);(1,3,5,14,6,15,20,21)(2,7,19,12,9,11,10,23)(4,13,18,28,17,27,30,31)(8,22,29,26,16,25,24,32);(1,2)(3,12,15,23)(4,8)(5,10)(6,9)(7,21,11,14)(13,26,27,32)(16,17)(18,24)(19,20)(22,31,25,28)(29,30);(1,4)(2,8)(3,13)(5,18)(6,17)(7,22)(9,16)(10,24)(11,25)(12,26)(14,28)(15,27)(19,29)(20,30)(21,31)(23,32)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=fh,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72240442, #72240447, #72240453, … |
| maximal normal subgroup of | #72240442, #72240447, #72240453, … |
| maximal quotient of | #72255845, #72255973 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282386);