Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356901 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,10)(4,8)(5,15)(6,9)(7,14)(11,16)(12,13)(18,22)(19,25)(20,29)(24,31)(27,28)(30,32);(1,3,5,14,6,11,4,13)(2,7,8,10,9,12,15,16)(17,18,20,22,21,24,29,31)(19,26,27,32,28,23,25,30);(3,11)(7,12)(10,16)(13,14)(17,19)(18,23)(20,27)(21,28)(22,30)(24,26)(25,29)(31,32);(1,2)(3,12)(4,15)(5,8)(6,9)(7,11)(10,13)(14,16)(18,24)(22,31)(23,26)(30,32);(1,4,6,5)(2,8,9,15)(3,13,11,14)(7,10,12,16)Polycyclic presentation:
2^9|b^2=f,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ehi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72338417, #72338418, #72338448, … |
| maximal normal subgroup of | #72338417, #72338418, #72338448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264102);