Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34727 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2104 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)(5,6)(7,10)(8,11)(9,12)(13,14,19,21,18,17,16,25)(15,20,22,28,24,23,26,27);(5,7,10,6)(8,11,9,12)(13,15)(14,20)(16,26)(17,23)(18,24)(19,22)(21,28)(25,27);(1,2,3,4)(13,16,18,19)(14,21,17,25)(15,22,24,26)(20,27,23,28);(5,8,10,9)(6,11,7,12);(5,9,10,8)(6,11,7,12)(13,17,19,25,18,14,16,21)(15,23,22,27,24,20,26,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=df,e^c=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72283957, #72284030, #72284153, … |
| maximal normal subgroup of | #72283957, #72284030, #72284153, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6311819);