Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454974 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#2017 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,22)(9,14)(10,15)(11,16)(12,19)(13,25)(18,33)(20,29)(21,28)(23,36)(24,32)(26,31)(27,35)(30,34);(5,7,9,19,11,21,10,20)(6,12,15,17,16,29,14,28)(8,23,25,35,26,34,24,18)(13,30,22,27,32,36,31,33);(5,8,10,25,11,26,9,24)(6,13,14,22,16,32,15,31)(7,18,19,34,21,35,20,23)(12,27,17,30,29,33,28,36);(2,4)(6,14,16,15)(8,24,26,25)(12,28,29,17)(18,34,35,23)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=eg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376963);