Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470402 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1572 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)(17,18)(19,26)(20,21)(22,24)(23,28)(25,27)(29,30)(31,32);(9,11)(10,13)(12,15)(14,16)(17,19,27,31,22,29,21,28)(18,23,20,30,24,32,25,26);(1,3,4,8)(2,5,6,7)(11,15)(13,16)(17,20)(18,21)(19,26)(22,25)(23,31)(24,27)(28,32)(29,30);(11,15)(13,16)(17,18)(19,23)(20,27)(21,25)(22,24)(26,28)(29,32)(30,31);(10,14)(13,16)(18,24)(20,25)(23,32)(26,30)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6382628);