Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483956 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
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 | C2×#493 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
641 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)(3,6)(4,5)(7,8)(10,16,19,18)(11,21,20,22)(12,17)(13,24)(14,23);(2,5)(6,8)(9,10,13,18,14,20,24,22,15,19,12,16,23,11,17,21);(1,3,4,7)(2,6,5,8)(9,11,12,22,14,10,17,16,15,20,13,21,23,19,24,18);(10,11,19,20)(12,13)(16,21,18,22)(17,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,d^2=g,e^2=h,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,d^b=dh,g^b=gi,d^c=dh,g^c=gi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224853, #72225294, #72225297, … |
| maximal normal subgroup of | #72224853, #72225294, #72225297, … |
| maximal quotient of | #72224828, #72309669, C2×#496059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403255);