Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9385 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1437 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,9,5,10,4,7)(3,11,14,15,12,8,13,16)(17,18)(19,22,26,30)(20,23)(21,24)(25,31,27,29)(28,32);(2,7,10,9)(3,12)(8,16,11,15)(13,14)(17,19)(18,22,24,30)(20,27)(21,26)(23,31,32,29)(25,28);(1,3,4,13,5,12,6,14)(2,8,7,15,10,11,9,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,32)(26,27)(30,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,f^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=gh,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17789);