Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2358 |
| prime radical | C2×#453 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
802 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,4,3,6,8)(9,10)(11,19)(12,16)(13,20)(14,18)(15,22)(17,24)(21,23);(1,3,5,8,4,2,6,7)(9,11)(10,12)(13,21)(14,15)(16,24)(17,22)(18,23)(19,20);(2,3)(7,8)(9,12,14,23)(10,15,18,11)(13,19,24,22)(16,17,21,20);(9,12,14,23)(10,16,18,21)(11,20,15,17)(13,22,24,19)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ci,d^b=de,e^b=eh,f^c=fh,e^d=eh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 134365);