Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,11)(8,10)(9,12)(14,17)(15,19)(21,22)(23,27)(24,26)(25,28);(5,6,8,10)(7,9,12,11)(21,22,24,26)(23,25,28,27);(1,2,4,3)(13,14)(15,18)(16,17)(19,20)(21,22)(23,25)(24,26)(27,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,23,24,28)(22,25,26,27);(1,3,4,2)(5,6)(7,11)(8,10)(9,12)(13,15,16,19)(14,18,17,20)(21,22)(23,27)(24,26)(25,28)Polycyclic presentation:
2^9|b^2=f,c^2=fgi,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1372814);