Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2782 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
595 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)(7,18)(8,25)(9,13)(10,16)(11,17)(12,22)(14,28)(15,19)(20,29)(21,34)(23,31)(24,30)(26,32)(27,33)(35,36);(5,7,19,22,11,24,10,23)(6,12,15,30,17,31,16,18)(8,26,28,35,27,21,9,20)(13,32,25,36,14,34,33,29);(1,3,4,2)(5,8,10,9,11,27,19,28)(6,13,15,25,17,14,16,33)(7,20,22,26,24,35,23,21)(12,29,18,34,31,36,30,32);(5,9,10,27,11,28,19,8)(6,14,16,25,17,13,15,33)(7,21,22,20,24,26,23,35)(12,29,30,32,31,36,18,34)Polycyclic presentation:
2^9|a^2=h,b^2=efg,c^2=eh,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298327);