Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6,9,10)(7,11,8,12)(13,14)(15,19)(16,27)(17,21)(18,22)(20,29)(23,35)(24,38)(25,34)(26,33)(28,32)(30,41)(31,37)(36,42)(39,43)(40,44);(2,3)(6,10)(11,12)(13,15,17,25)(14,19,21,34)(16,28,29,23)(18,24,31,26)(20,35,27,32)(22,38,37,33)(30,40,42,39)(36,43,41,44);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,17,29)(14,20,21,27)(15,23,25,28)(18,30,31,42)(19,32,34,35)(22,36,37,41)(24,39,26,40)(33,43,38,44);(1,3,4,2)(5,7,9,8)(6,11,10,12)(14,21)(15,24)(18,31)(19,33)(20,27)(23,39)(25,26)(28,40)(30,42)(32,43)(34,38)(35,44);(5,8,9,7)(6,11,10,12)(15,25)(16,29)(19,34)(20,27)(24,26)(30,42)(33,38)(36,41)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=fgi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7473184);