Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #37681 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10)(11,17)(12,15)(13,22)(14,20)(16,21)(18,23)(19,24);(1,2,3,4)(5,6)(7,8)(9,10,12,15)(11,18,19,14)(13,16,21,22)(17,20,24,23);(6,8)(9,11,12,19)(10,14,15,18)(13,23,21,20)(16,24,22,17);(5,6)(7,8)(9,10)(11,18)(12,15)(13,16)(14,19)(17,23)(20,24)(21,22);(6,8)(10,15)(14,18)(16,22)(17,24)Polycyclic presentation:
2^9|b^2=f,c^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, 6481578);