Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #36687 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9,12,16)(4,8)(5,7)(6,11,15,13)(10,14)(17,18,23,26,22,19,21,25)(20,24,30,32,28,27,29,31);(1,3,10,11,5,12,4,13)(2,6,14,9,7,15,8,16)(17,19,22,18)(20,27,28,24)(21,25,23,26)(29,31,30,32);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,20)(18,24)(19,27)(21,30)(22,28)(23,29)(25,32)(26,31);(2,7)(6,15)(8,14)(9,16)(20,28)(24,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102207);