Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×D4×D8 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,9,11,8,7,12,10)(13,14)(15,18)(16,25)(17,21)(19,22)(20,23)(24,27)(26,28);(2,4)(9,12)(10,11)(14,18)(15,22)(16,23)(19,20)(21,27)(24,28)(25,26);(5,7)(6,8)(9,10)(11,12)(14,18)(15,22)(16,23)(19,20)(21,27)(24,28)(25,26)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=gi,f^2=ghi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dh,e^b=eg,f^b=fghi,g^b=gi,h^b=hi,d^c=dg,e^c=eg,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225208, D8×#357, #72226890 |
| maximal normal subgroup of | #72225208, D8×#357, #72226890 |
| maximal quotient of | #72225182, #72226864, #72226885, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39044);