Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,10)(11,12)(13,14,17,16)(15,18,20,19)(23,27)(25,28);(5,6)(7,12)(8,11)(9,10)(14,16)(18,19)(21,22,26,24)(23,28,27,25);(5,7,9,8)(6,11,10,12)(13,15,17,20)(14,18,16,19)(21,23)(22,25)(24,28)(26,27);(1,2)(3,4)(5,8,9,7)(6,12,10,11)(13,16,17,14)(15,19,20,18)(21,24,26,22)(23,28,27,25);(5,6)(7,12)(8,11)(9,10)(13,15)(14,19)(16,18)(17,20)(21,25,26,28)(22,27,24,23)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=f,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4030497);