Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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 | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2053 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14,17,21)(15,22,18,19)(16,20,24,28)(23,27,25,26);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,17,18)(14,19,21,22)(16,23,24,25)(20,26,28,27);(2,4)(5,7,9,8)(6,11,12,10)(14,19)(15,18)(16,24)(20,27)(21,22)(26,28);(1,2)(3,4)(5,8,9,7)(6,10,12,11)(13,16,17,24)(14,20,21,28)(15,23,18,25)(19,26,22,27);(2,4)(16,24)(20,28)(23,25)(26,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,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, 6484778);