Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,4)(6,7)(9,10)(11,15)(12,16)(13,17)(14,19)(18,22)(20,24)(21,23);(5,6,8,7)(10,15,19,17)(12,21)(16,18,23,24)(20,22);(5,7,8,6)(10,16)(11,13)(12,22)(15,18)(17,24)(19,23)(20,21)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=dh,d^2=i,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,g^a=gi,h^a=hi,e^b=ei,d^c=di,e^c=eh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216446, #72216458, #72216468, … |
| maximal normal subgroup of | #72216446, #72216458, #72216468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54103);