Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573477 |
| 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 | #2867 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,10,11,12)(13,14,19,18)(15,20,25,28)(16,21,17,22)(23,27,24,26);(6,9)(10,12)(13,15,19,25)(14,20,18,28)(16,26,17,27)(21,23,22,24);(1,2,3,4)(13,16,19,17)(14,21,18,22)(15,23,25,24)(20,27,28,26);(5,7,8,11)(6,10,9,12)(14,18)(16,21)(17,22)(20,28)(23,26)(24,27);(13,17,19,16)(14,22,18,21)(15,24,25,23)(20,26,28,27)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=gi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6490856);