Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| 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 | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,11)(7,10,8,12)(13,14,18,25,20,26,19,24)(15,27,31,39,32,38,30,21)(16,22,17,23,34,41,33,40)(28,43,44,42,35,37,29,36);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,21)(16,29)(17,35)(18,30)(19,31)(20,32)(22,37)(23,42)(24,27)(25,38)(26,39)(28,33)(34,44)(36,40)(41,43);(2,4)(5,8,9,7)(6,10,11,12)(13,16,19,33,20,34,18,17)(14,22,24,40,26,41,25,23)(15,28,30,29,32,35,31,44)(21,36,38,37,39,42,27,43);(2,4)(13,17,18,34,20,33,19,16)(14,23,25,41,26,40,24,22)(15,29,31,28,32,44,30,35)(21,37,27,36,39,43,38,42);(13,18,20,19)(14,24,26,25)(15,30,32,31)(16,17,34,33)(21,27,39,38)(22,40,41,23)(28,29,35,44)(36,43,42,37)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=fgi,d^2=f,e^2=i,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=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355308);