Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#392 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2514 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)(6,10)(11,12)(13,14,17,21,18,15,19,23)(16,20,25,27,24,22,26,28);(5,6,9,10)(7,11,8,12)(13,15,17,23,18,14,19,21)(16,22,25,28,24,20,26,27);(2,4)(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28);(5,8,9,7)(6,11,10,12)(16,24)(20,22)(25,26)(27,28);(14,15)(20,22)(21,23)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=f,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=df,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72284044, #72284075, #72284285, … |
| maximal normal subgroup of | #72284044, #72284075, #72284285, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289345);