Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1854 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)(7,11)(8,12)(9,10)(13,14,19,20)(15,23,21,17)(16,18,22,28)(24,25,26,27);(6,10)(11,12)(13,15,19,21)(14,17,20,23)(16,25,22,27)(18,24,28,26);(1,2,3,4)(13,16)(14,18)(15,24)(17,27)(19,22)(20,28)(21,26)(23,25);(5,7,9,8)(6,11,10,12)(13,17,19,23)(14,21,20,15)(16,26,22,24)(18,25,28,27);(5,8,9,7)(6,12,10,11)(13,18,19,28)(14,22,20,16)(15,25,21,27)(17,24,23,26)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6492149);