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 | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14517 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1033 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,3,12,10)(4,15,18,13,20,16,19,8)(5,14,23,21,24,9,22,26)(17,28,31,27,32,29,30,25);(1,3)(2,7)(4,16)(5,21)(6,10)(8,18)(9,23)(11,12)(13,19)(14,22)(15,20)(17,25)(24,26)(27,32)(28,31)(29,30);(1,4,6,18,7,20,12,19)(2,8,11,15,3,13,10,16)(5,17,23,31,24,32,22,30)(9,25,26,28,14,27,21,29);(1,5)(2,9)(3,14)(4,17)(6,23)(7,24)(8,25)(10,21)(11,26)(12,22)(13,27)(15,28)(16,29)(18,31)(19,30)(20,32)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=i,f^2=i,g^2=i,h^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, 295390);