Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| 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×#532 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2404 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)(8,9)(11,12)(13,14,18,21,17,20,26,28)(15,16,23,25,22,24,19,27);(6,9)(10,12)(13,15,18,23,17,22,26,19)(14,16,21,25,20,24,28,27);(2,4)(13,16)(14,19)(15,21)(17,24)(18,25)(20,23)(22,28)(26,27);(5,7,8,11)(6,10,9,12)(14,20)(15,22)(19,23)(21,28);(14,20)(16,24)(21,28)(25,27)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,d^b=di,e^c=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72307448, #72307540, #72307695 |
| maximal normal subgroup of | #72307448, #72307540, #72307695 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278941);