Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12721 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1782 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1782 |
| prime radical | #62815 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2217 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)(3,14,17,28)(4,11,20,25)(5,10)(6,12)(7,13)(8,15)(9,19,26,18)(16,22,21,32)(23,29)(24,30)(27,31);(1,3)(2,8)(4,15)(5,16)(6,21)(7,17)(9,22)(10,23)(11,14)(12,27)(13,24)(18,29)(19,31)(20,30)(25,28)(26,32);(1,4,7,20)(2,9)(3,15,17,30)(5,18,6,19)(8,22)(10,25)(11,12)(13,26)(14,27)(16,29,21,31)(23,28)(24,32);(1,5,7,6)(2,10,13,12)(3,16,17,21)(4,18,20,19)(8,23,24,27)(9,25,26,11)(14,22,28,32)(15,29,30,31)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,f^b=fi,g^b=gh,e^c=ei,g^c=gi,g^e=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400646);