Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| 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 | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1826 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)(13,14)(15,24)(16,27)(17,22)(18,23)(19,21)(20,26)(25,28);(1,2,3,4)(6,10)(8,11)(13,15,19,16)(14,20,21,28)(17,25,18,26)(22,24,23,27);(5,6)(7,8)(9,10)(11,12)(13,16,19,15)(14,20,21,28)(17,26,18,25)(22,24,23,27);(5,7,9,12)(6,11,10,8)(14,21)(20,28)(22,23)(24,27);(5,8,9,11)(6,7,10,12)(13,17,19,18)(14,22,21,23)(15,25,16,26)(20,24,28,27)Polycyclic presentation:
2^9|b^2=fgi,c^2=i,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6528268);