Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18376 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #34937 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1385 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,7,5,3,4,8)(9,10)(11,20,22,23)(12,17,16,21)(13,24)(14,18)(15,19);(1,3,5,2)(4,8,6,7)(9,11,13,21,15,22,14,17)(10,16,24,20,19,12,18,23);(1,4,5,6)(2,7,3,8)(9,12,15,16)(10,17,19,21)(11,18,22,24)(13,23,14,20);(1,4,5,6)(2,8,3,7)(9,13,15,14)(10,18,19,24)(11,17,22,21)(12,23,16,20)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=h,d^2=h,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,g^c=gi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102247);