Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| 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 | #36455 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2216 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)(7,12)(8,10)(9,11)(13,14,20,19)(15,17,22,28)(16,21,18,23)(24,27,25,26);(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,26)(18,27)(19,28)(20,22)(21,24)(23,25);(1,2,3,4)(13,16,20,18)(14,21,19,23)(15,24,22,25)(17,27,28,26);(5,8,9,7)(6,10,11,12)(13,17,20,28)(14,22,19,15)(16,24,18,25)(21,27,23,26);(13,18,20,16)(14,23,19,21)(15,25,22,24)(17,26,28,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491083);