Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14757 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8)(3,5,4,7)(9,10,14,15)(11,12,16,13)(17,18,20,22)(19,23,24,21)(27,31)(30,32);(1,3,6,4)(2,7,8,5)(9,11,14,16)(10,13,15,12)(25,26,29,28)(27,30,31,32);(1,3,6,4)(2,5,8,7)(9,12,14,13)(10,11,15,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,30)(28,32)(29,31);(1,4,6,3)(2,5,8,7)(9,13,14,12)(10,11,15,16)(17,19,20,24)(18,21,22,23)(25,28,29,26)(27,32,31,30);(1,5)(2,4)(3,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=f,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72303399, #72303502, #72303510, … |
| maximal normal subgroup of | #72303399, #72303502, #72303510, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1359241);