Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #199806 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60750 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1303 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(3,10)(4,7)(9,14)(11,12)(13,16)(17,18)(19,22,26,29)(20,23)(21,24)(25,28,31,32)(27,30);(1,2,7,14,5,9,4,6)(3,8,10,16,11,15,12,13)(17,19,21,26)(18,22)(20,25,27,31)(23,28)(24,29)(30,32);(1,3)(2,8)(4,12)(5,11)(6,13)(7,10)(9,15)(14,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32);(3,11)(8,15)(10,12)(13,16)(20,27)(23,30)(25,31)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,g^b=gi,d^c=dh,f^c=fhi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107003);