Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8346 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#379 |
| prime radical | #62815 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,6,3,4,8)(9,10)(11,18)(12,15)(13,17)(14,22)(16,20)(19,23)(21,24);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,19)(13,20)(14,23)(15,21)(17,24)(18,22);(1,4,6,5)(2,7,3,8)(9,12,15,10)(11,19,21,16)(13,22,14,17)(18,23,24,20);(9,13,15,14)(10,17,12,22)(11,20,21,23)(16,24,19,18)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=h,e^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92575);