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 | C22×#127 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2753 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)(3,6)(4,5)(7,8)(10,12)(11,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,32)(29,31);(2,5)(6,7)(9,10)(11,15)(12,13)(14,16)(17,19,20,24)(18,21,23,22)(25,26,29,31)(27,32,28,30);(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,27,29,28)(26,30,31,32);(1,3,4,8)(2,6,5,7)(9,11)(10,15)(12,14)(13,16)(17,18)(19,21)(20,23)(22,24)(25,28,29,27)(26,30,31,32);(1,2)(3,7)(4,5)(6,8)(9,12,13,10)(11,15,16,14)(17,19)(18,22)(20,24)(21,23)(25,26)(27,30)(28,32)(29,31)Polycyclic presentation:
2^9|b^2=gh,c^2=gi,d^2=fghi,e^2=g,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 | #72303442, #72303520, #72303524, … |
| maximal normal subgroup of | #72303442, #72303520, #72303524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1406756);