Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36102 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(18,19)(22,23);(2,6)(7,8)(18,22)(19,23)(21,24);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,18,20,19)(21,23,24,22);(9,11,12,15)(10,14,13,16)(18,23)(19,22)(21,24);(1,4,5,3)(2,8,6,7)(17,19,20,18)(21,22,24,23)Polycyclic presentation:
2^9|c^2=fi,d^2=gi,e^2=fi,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.
| maximal subgroup of | #72305296, #72305305, #72305307, … |
| maximal normal subgroup of | #72305296, #72305305, #72305307, … |
| maximal quotient of | #72305897, #72306005, #72315881 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6494987);