Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| 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 | #34977 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,19,23,18,24,17,22)(15,25,16,21,27,20,26,28);(5,7,10,6)(8,11,9,12)(13,15,17,26,18,27,19,16)(14,20,22,21,24,25,23,28);(2,4)(13,16,19,27,18,26,17,15)(14,21,23,20,24,28,22,25);(5,8,10,9)(6,11,7,12)(13,17,18,19)(14,22,24,23)(15,26,27,16)(20,21,25,28);(5,9,10,8)(6,11,7,12)(13,17,18,19)(14,23,24,22)(15,26,27,16)(20,28,25,21)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=h,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6302848);