Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(6,9)(10,12)(13,14,17,15)(16,19,20,18)(21,22)(23,27)(24,25)(26,28);(2,3)(13,14)(15,17)(16,18)(19,20)(21,23,26,24)(22,27,28,25);(2,3)(14,15)(18,19)(22,28)(23,24);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18)(21,24,26,23)(22,25,28,27);(5,7,8,11)(6,10,9,12)(13,16,17,20)(14,18,15,19)(21,25)(22,23)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,d^2=g,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232932, #72232935, #72232944, … |
| maximal normal subgroup of | #72232932, #72232935, #72232944, … |
| maximal quotient of | #72328794, #72330943, #72331023 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4838679);