Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9345 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #3006 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1239 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,4,7)(3,11,13,8,14,15,12,16)(17,18)(19,23,29,32)(20,24)(21,25)(22,26)(27,30,28,31);(2,7,10,9)(8,11,16,15)(17,19)(18,23,26,32)(20,27)(21,28)(22,29)(24,30,25,31);(1,3,6,14)(2,8,10,16)(4,12,5,13)(7,11,9,15)(17,20,22,21)(18,24,26,25)(19,27,29,28)(23,30,32,31);(17,21,22,20)(18,25,26,24)(19,28,29,27)(23,31,32,30)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=i,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,g^b=gi,f^c=fhi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107229);