Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533105 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24152 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1666 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)(3,7)(4,8)(5,6)(9,10,12,15,11,13,16,14)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32);(2,6)(7,8)(10,13)(14,15)(18,22)(19,25)(24,29)(26,31);(1,3,5,4)(2,7,6,8)(10,14)(12,16)(13,15)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32);(1,4,5,3)(2,8,6,7)(20,27)(24,29)(26,31)(30,32);(10,13)(14,15)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)Polycyclic presentation:
2^9|a^2=g,c^2=fi,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72313265, #72313283, #72313318, … |
| maximal normal subgroup of | #72313265, #72313283, #72313318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6457779);