Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4005 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,10)(7,11,12,8)(14,17)(18,20)(22,24)(23,27);(6,10)(7,12)(13,14,16,17)(15,18,19,20)(21,22,25,24)(23,28,27,26);(6,10)(8,11)(15,19)(18,20)(22,24)(26,28);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,28)(25,27);(5,8)(6,7)(9,11)(10,12)(13,15,16,19)(14,18,17,20)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^9|a^2=g,b^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72281140, #72281583, #72281715, … |
| maximal normal subgroup of | #72281140, #72281583, #72281715, … |
| maximal quotient of | #72328642, #72329188, #72329979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4287278);