Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3062 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3337 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14)(15,24)(16,21)(17,22)(18,27)(19,23)(20,25)(26,28);(1,2)(3,4)(13,15,18,25,19,26,16,22)(14,20,21,24,23,17,27,28);(2,4)(5,7,9,8)(6,10,11,12)(13,16,19,18)(14,21,23,27)(15,25,26,22)(17,24,20,28);(2,4)(5,8,9,7)(6,10,11,12)(13,17)(14,22)(15,21)(16,24)(18,28)(19,20)(23,25)(26,27);(15,26)(17,20)(22,25)(24,28)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359293);