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 | #36259 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2122 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)(13,14,17,20)(15,16,23,19)(18,21,27,28)(22,25,24,26);(5,7,9,8)(6,10,11,12)(13,15,17,23)(14,19,20,16)(18,22,27,24)(21,26,28,25);(1,2,3,4)(15,22)(16,25)(18,27)(19,26)(21,28)(23,24);(5,8,9,7)(6,10,11,12)(13,16)(14,15)(17,19)(18,25)(20,23)(21,22)(24,28)(26,27);(15,22)(16,25)(18,27)(19,26)(21,28)(23,24)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=fi,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, 6491093);