Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2253 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(18,22)(19,23)(24,28)(25,30)(26,29)(31,32);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,18,21,22)(19,23,26,29)(20,24,27,28)(25,30,31,32);(11,15)(13,16)(17,19,21,26)(18,23,22,29)(20,25,27,31)(24,30,28,32);(11,15)(13,16)(17,20)(18,24)(19,25)(21,27)(22,28)(23,30)(26,31)(29,32);(10,14)(13,16)(20,27)(24,28)(25,31)(30,32)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377687);