Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 | |
4079 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)(7,11)(8,12)(9,10)(13,14)(15,19)(16,18)(17,20)(22,26)(24,27);(6,10)(8,11)(13,14,16,18)(15,17,20,19)(21,22,25,26)(23,24,28,27);(6,10)(7,12)(15,20)(17,19)(21,23,25,28)(22,27,26,24);(1,2)(3,4)(5,7,9,12)(6,11,10,8)(13,15)(14,17)(16,20)(18,19)(21,23)(22,24)(25,28)(26,27);(5,8,9,11)(6,12,10,7)(13,14)(15,17)(16,18)(19,20)(21,24)(22,23)(25,27)(26,28)Polycyclic presentation:
2^9|b^2=fg,c^2=h,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72234811, #72234816, #72234828, … |
| maximal normal subgroup of | #72234811, #72234816, #72234828, … |
| maximal quotient of | #72328642, #72330583, #72330748 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4220488);