Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,11,8,7,10,12)(13,14,17,16,18,20,15,19);(5,7,9,12,8,6,10,11)(14,16)(15,17)(19,20);(2,4)(6,7)(11,12)(13,15,18,17)(14,19,20,16);(13,16,15,14,18,19,17,20)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,c^b=ci,d^b=de,e^b=eh,g^b=gh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 136319);