Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8346 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6,5,8,3,7)(9,10)(11,18)(12,21)(13,16)(14,17)(15,19)(20,24)(22,23);(2,6)(3,4)(7,8)(9,11,13,19,14,20,12,22)(10,15,16,24,17,23,21,18);(9,10)(11,15)(12,21)(13,16)(14,17)(18,22)(19,24)(20,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=di,e^b=eh,f^b=fhi,d^c=dh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72279713, #72279735, #72279851, … |
| maximal normal subgroup of | #72279713, #72279735, #72279851, … |
| maximal quotient of | #72318976 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13816);