Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#3489 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(13,14)(15,24)(16,27)(17,21)(18,28)(19,23)(20,26)(22,25);(1,3,4,2)(6,9)(10,11)(13,15,19,26)(14,20,23,24)(16,25,21,18)(17,22,27,28);(5,6)(7,10)(8,9)(11,12)(13,14,19,23)(15,24,26,20)(16,17,21,27)(18,22,25,28);(5,7)(6,10)(8,12)(9,11)(13,16)(14,17)(15,25)(18,26)(19,21)(20,22)(23,27)(24,28);(13,17)(14,21)(15,22)(16,23)(18,20)(19,27)(24,25)(26,28);(13,18,19,25)(14,22,23,28)(15,16,26,21)(17,20,27,24)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533956);