Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533105 |
| 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 | C2×#3498 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(13,14,16,18)(15,17,20,19);(5,6,10,9)(7,12,8,11)(14,17)(15,20)(18,19);(2,4)(5,7,10,8)(6,11,9,12);(5,8,10,7)(6,11,9,12)(14,18)(17,19);(5,9,10,6)(7,12,8,11)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,f^b=fi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72283957, #72284723, #72285021 |
| maximal normal subgroup of | #72283957, #72284723, #72285021 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6479440);