Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #272923 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11,8,10)(9,12)(13,14)(15,22,24,28)(16,21)(17,19)(18,23,20,25)(26,27);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,18)(16,25)(17,24)(19,20)(21,28)(22,27)(23,26);(2,4)(5,6)(7,10)(8,11)(9,12)(14,19)(16,26)(18,20)(23,25);(5,8,9,7)(6,11,12,10)(13,15)(14,20)(16,25)(17,24)(18,19)(21,22)(23,26)(27,28)Polycyclic presentation:
2^9|a^2=fi,b^2=h,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,e^d=ei,f^d=fhi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 180200);