Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2580 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,11,15)(7,16,20,32)(8,13,24,30)(9,25,23,36)(10,26,22,14)(12,27,29,19)(17,33,21,31)(18,34,35,28);(5,7,11,20)(6,12)(8,21,24,17)(9,18,23,35)(10,19,22,27)(13,28)(14,16)(15,29)(25,31)(26,32)(30,34)(33,36);(5,8,22,9,11,24,10,23)(6,13,14,25,15,30,26,36)(7,17,27,35,20,21,19,18)(12,28,16,31,29,34,32,33);(1,3,4,2)(5,9,22,24,11,23,10,8)(6,13,26,36,15,30,14,25)(7,18,27,17,20,35,19,21)(12,28,32,33,29,34,16,31)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=eg,d^2=eh,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286710);