Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #298795 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,11)(7,12)(8,9)(13,14)(15,23)(16,19)(17,26)(18,22)(20,24)(21,27)(25,28);(1,2)(3,4)(5,6,9,7,10,12,8,11)(13,15,17,16,18,24,27,25)(14,19,21,23,22,28,26,20);(2,4)(5,7,9,12,10,11,8,6);(5,6,8,11,10,12,9,7)(13,16,17,24,18,25,27,15)(14,20,21,19,22,23,26,28)Polycyclic presentation:
2^9|b^2=ei,c^2=f,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 206001);