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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3042 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)(15,16)(17,19)(18,20)(22,24)(26,27);(13,15,17,20)(14,18,19,16)(21,22,25,24)(23,27,28,26);(1,2)(3,4)(14,19)(16,18)(22,24)(26,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19)(21,23)(22,26)(24,27)(25,28);(5,7,8,11)(6,10,9,12)(13,16)(14,15)(17,18)(19,20)(21,24,25,22)(23,27,28,26)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,e^2=fgi,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4306417);