Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3444 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,21)(18,22)(23,26)(24,27)(25,28);(2,4)(13,15,18,25)(14,19,22,28)(16,24,17,23)(20,27,21,26);(6,9)(10,12)(13,16,18,17)(14,20,22,21)(15,23,25,24)(19,26,28,27);(13,17,18,16)(14,21,22,20)(15,24,25,23)(19,27,28,26);(14,22)(19,28)(20,21)(26,27);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,f^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7535706);