Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| 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×#3036 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2304 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)(13,14,18,19,17,22,16,23)(15,21,24,28,25,20,26,27);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(14,19)(15,24)(16,18)(20,21)(22,23)(25,26);(2,4)(5,8,9,7)(6,10,11,12)(13,15,16,26,17,25,18,24)(14,20,23,28,22,21,19,27);(2,4)(13,15,16,26,17,25,18,24)(14,21,23,27,22,20,19,28);(14,22)(15,25)(19,23)(24,26)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=fgi,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355338);