Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470874 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25117 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,26)(20,23)(21,25)(22,30)(24,27)(28,31)(29,32);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,19,27,28,21,29,20,30)(18,22,23,32,25,31,24,26);(11,15)(13,16)(17,20,21,27)(18,23,25,24)(19,28,29,30)(22,26,31,32);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(19,29)(22,31)(26,32)(28,30)Polycyclic presentation:
2^9|c^2=fgi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6382129);