Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483873 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10,13,17,14,19,23,18)(11,16,20,15,21,24,12,22);(1,3,5,4)(2,6,7,8);(10,15)(11,20)(12,21)(13,23)(16,17)(18,24)(19,22);(1,4,5,3)(2,6,7,8)(9,11,13,20,14,21,23,12)(10,16,17,15,19,24,18,22)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=eg,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=dh,f^c=fhi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216494, C2×#496007, #72239540, … |
| maximal normal subgroup of | #72216494, C2×#496007, #72239540, … |
| maximal quotient of | #72216458, C2×#496007, #72309683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403203);