Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1427 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)(7,16)(8,21)(9,26)(10,15)(11,14)(12,19)(13,24)(17,34)(18,28)(20,32)(22,33)(23,30)(25,31)(27,36)(29,35);(5,7,11,20)(6,12,14,28)(8,22,25,35)(9,18,10,19)(13,29,30,33)(15,32,26,16)(17,23,36,24)(21,34,31,27);(5,8)(6,13,14,30)(7,17)(9,24)(10,23)(11,25)(12,27,28,34)(15,21,26,31)(16,33,32,29)(18,35)(19,22)(20,36);(6,14)(7,18)(9,10)(12,16)(13,30)(17,35)(19,20)(22,36)(23,24)(27,33)(28,32)(29,34)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=eg,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,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293125);