Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6358057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1968 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(4,7)(8,12)(9,13)(11,14)(15,16)(19,25)(20,27)(22,28)(24,30)(26,31)(29,32);(1,2,7,12,5,8,4,6)(3,9,14,16,10,15,11,13)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(2,8)(3,10)(6,12)(11,14)(17,19,20,25,21,26,27,31)(18,22,24,28,23,29,30,32);(1,3)(2,9)(4,11)(5,10)(6,13)(7,14)(8,15)(12,16)(19,26)(22,29)(25,31)(28,32);(3,10)(9,15)(11,14)(13,16)(18,23)(22,29)(24,30)(28,32)Polycyclic presentation:
2^9|b^2=fh,c^2=g,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,e^b=ei,d^c=dhi,e^d=eh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72339707, #72339799, #72339836, … |
| maximal normal subgroup of | #72339707, #72339799, #72339836, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265255);