Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,11,12,9)(14,19)(15,23)(16,18)(20,27)(24,25)(26,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,14)(15,20)(16,21)(17,19)(18,22)(23,28)(24,26)(25,27);(2,4)(13,15,17,24)(14,20,19,26)(16,23,18,25)(21,28,22,27);(1,2)(3,4)(14,19)(20,26)(21,22)(27,28);(2,4)(13,16,17,18)(14,21,19,22)(15,23,24,25)(20,28,26,27)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481971);