Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| 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 | #35238 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,20,27,19,23,18,24)(15,25,16,21,26,28,17,22);(1,2)(3,4)(13,15,19,26)(14,21,23,22)(16,20,17,18)(24,28,27,25);(2,4)(6,9)(10,12)(13,16,18,15,19,17,20,26)(14,22,24,28,23,21,27,25);(2,4)(13,17,18,26,19,16,20,15)(14,22,24,28,23,21,27,25);(5,7,8,11)(6,10,9,12)(14,23)(21,22)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=f,d^2=f,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363686);