Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190124 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1524 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)(3,11,14,7)(4,15,17,21)(5,9)(6,10)(8,16,22,32)(12,26,29,24)(13,25,30,20)(18,23)(19,28,31,27);(1,3)(2,7)(4,12)(5,14)(6,13)(8,19)(9,11)(10,20)(15,31)(16,28)(17,27)(18,30)(21,24)(22,26)(23,25)(29,32);(1,4)(2,8,9,21)(3,12)(5,16)(6,17)(7,19,11,24)(10,22,23,15)(13,27)(14,28)(18,32)(20,26,25,31)(29,30);(3,13)(7,20)(11,25)(12,27)(14,30)(19,26)(24,31)(28,29)Polycyclic presentation:
2^9|a^2=g,c^2=e,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,g^b=gh,f^c=fh,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72240651, #72240652, #72251351, … |
| maximal normal subgroup of | #72240651, #72240652, #72251351, … |
| maximal quotient of | #72255653 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97320);