Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,11)(7,10,8,12)(13,14,18,17,20,23,19,22)(15,24,16,21,26,27,25,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,20,26)(14,21,23,28)(16,18,25,19)(17,24,22,27);(2,4)(5,8,9,7)(6,10,11,12)(13,16,19,15,20,25,18,26)(14,21,22,24,23,28,17,27);(2,4)(13,17,19,14,20,22,18,23)(15,21,25,24,26,28,16,27);(13,18,20,19)(14,17,23,22)(15,25,26,16)(21,24,28,27)Polycyclic presentation:
2^9|a^2=fgi,b^2=gi,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355319);