Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| 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 | #37277 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1849 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(13,14,19,21)(15,22,24,17)(16,18,26,23)(20,27,28,25);(5,6)(7,10)(8,9)(11,12)(13,15,23,27,19,24,18,25)(14,20,26,17,21,28,16,22);(2,4)(6,9)(7,11)(13,16,18,21,19,26,23,14)(15,17,25,20,24,22,27,28);(5,7,8,11)(6,10,9,12)(13,17,19,22)(14,15,21,24)(16,25,26,27)(18,20,23,28);(6,9)(10,12)(15,24)(17,22)(20,28)(25,27)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6421716);