Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| 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×#3499 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8094 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)(5,6,9,7)(8,11,12,10)(14,18)(16,23)(19,24)(22,27);(6,7)(10,11)(15,21)(19,24)(22,27)(26,28);(1,3,4,2)(5,7,9,6)(8,11,12,10)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28);(5,8,9,12)(6,10,7,11);(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533486);