Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2051 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(7,11)(10,12)(13,14,20,25,19,15,18,22)(16,21,27,28,24,23,26,17);(6,9)(7,11)(13,15,19,14)(16,23,24,21)(17,27,28,26)(18,22,20,25);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,23)(17,22)(18,26)(19,24)(20,27)(25,28);(6,9)(10,12)(13,17,18,23,19,28,20,21)(14,16,22,26,15,24,25,27);(5,7,8,11)(6,10,9,12)(16,24)(17,28)(21,23)(26,27)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=g,d^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7063525);