Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62842 |
| 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×#3522 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,4,3)(6,9)(10,12)(13,14,20,17)(15,23,22,19)(16,25,18,21)(24,26,27,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,24)(17,23)(18,27)(20,22)(21,26)(25,28);(13,16,20,18)(14,21,17,25)(15,24,22,27)(19,26,23,28);(13,17,20,14)(15,23,22,19)(16,21,18,25)(24,26,27,28);(1,3,4,2)(13,18,20,16)(14,21,17,25)(15,24,22,27)(19,28,23,26);(5,7,8,11)(6,10,9,12)(13,19,20,23)(14,22,17,15)(16,26,18,28)(21,27,25,24)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fh,e^b=eh,d^c=dh164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532946);