Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
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 | |
1119 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)(5,6,11,16)(7,17,20,25)(8,14,23,29)(9,12,10,15)(13,24,28,18)(19,33,35,31)(21,30,22,26)(27,32,36,34);(6,12)(7,18)(8,21)(9,10)(14,29)(15,16)(17,25)(20,24)(22,23)(27,33)(31,36)(32,34);(1,3,4,2)(5,7,11,20)(6,13,16,28)(8,19,23,35)(9,18,10,24)(12,25,15,17)(14,27,29,36)(21,34,22,32)(26,33,30,31);(5,8)(6,14)(7,19)(9,22)(10,21)(11,23)(12,26)(13,27)(15,30)(16,29)(17,31)(18,32)(20,35)(24,34)(25,33)(28,36)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295388);