Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| 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×#3548 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7096 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)(7,11)(8,10)(9,12)(13,14)(15,17)(16,19)(18,24)(20,22)(21,26)(23,27)(25,28);(5,6,8,10)(7,9,12,11)(13,15)(14,20)(16,17)(18,23)(19,22)(21,27)(24,25)(26,28);(5,6)(7,9)(8,10)(11,12)(14,16)(15,22)(21,24)(23,28);(5,7)(6,9)(8,12)(10,11)(13,16,19,14)(15,17,22,20)(18,24,26,21)(23,25,28,27);(1,3,4,2)(13,17)(14,15)(16,22)(18,25)(19,20)(21,23)(24,28)(26,27);(13,18,19,26)(14,21,16,24)(15,23,22,28)(17,25,20,27)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533754);