Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2361 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)(6,8)(7,11)(14,18)(15,20)(21,22,24,26)(23,25,27,28);(7,11)(10,12)(13,14,17,18)(15,19,20,16)(21,23,24,27)(22,25,26,28);(5,6,9,8)(7,10,11,12)(13,15,17,20)(14,19,18,16)(21,23)(22,25)(24,27)(26,28);(1,2)(3,4)(5,7,9,11)(6,10,8,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,25)(24,26)(27,28);(5,8,9,6)(7,12,11,10)(13,15)(14,19)(16,18)(17,20)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3243907);