Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
933 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,17,11,16,31,15)(7,18,23,35,22,26,21,12)(8,24,27,36,28,13,29,20)(9,30,32,19,33,25,34,14);(5,7)(6,12)(8,25)(9,13)(10,23)(11,22)(14,27)(15,26)(16,35)(17,18)(19,29)(20,32)(21,31)(24,33)(28,30)(34,36);(1,2,4,3)(5,8,11,28)(6,13,16,24)(7,19,22,14)(9,26,33,18)(10,27,31,29)(12,34,35,32)(15,36,17,20)(21,30,23,25);(1,3,4,2)(5,9,31,34,11,33,10,32)(6,14,17,30,16,19,15,25)(7,20,23,24,22,36,21,13)(8,26,29,35,28,18,27,12)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292610);