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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2036 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)(5,6)(7,11)(8,10)(9,12)(13,14,19,20)(15,17,22,28)(16,23,18,21)(24,26,25,27);(5,7,9,8)(6,10,12,11)(13,15)(14,17)(16,26)(18,27)(19,22)(20,28)(21,24)(23,25);(2,4)(5,7,9,8)(6,11,12,10)(13,16,19,18)(14,21,20,23)(15,24,22,25)(17,27,28,26);(5,8,9,7)(6,10,12,11)(13,17,19,28)(14,22,20,15)(16,25,18,24)(21,26,23,27);(13,18,19,16)(14,23,20,21)(15,25,22,24)(17,26,28,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=ehi,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, 6484653);