Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2415 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9);(5,7,8,12)(6,9,10,11)(14,18)(15,16)(19,20);(2,4)(13,14)(15,18)(16,20)(17,19);(13,15)(14,19)(16,17);(13,14)(15,18)(16,20)(17,19)Polycyclic presentation:
2^9|a^2=f,b^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237160, C2×#6574610, #72249378, … |
| maximal normal subgroup of | #72237160, C2×#6574610, #72249378, … |
| maximal quotient of | #72237141, C2×#6574610, #72249747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481806);