Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1731 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)(5,6,9,8)(7,10,12,11)(13,14)(15,25)(16,29)(17,33)(18,35)(19,22)(20,28)(21,31)(23,34)(24,27)(26,40)(30,39)(32,38)(36,44)(37,42)(41,43);(2,3)(5,7)(6,10)(8,11)(9,12)(13,15,18,28,19,17,27,34)(14,20,24,25,22,23,35,33)(16,30,32,44,31,26,41,42)(21,37,38,40,29,36,43,39);(1,2,4,3)(5,8,9,6)(7,11,12,10)(13,16)(14,21)(15,26)(17,30)(18,32)(19,31)(20,36)(22,29)(23,37)(24,38)(25,39)(27,41)(28,42)(33,40)(34,44)(35,43);(1,3,4,2)(7,12)(10,11)(14,22)(16,31)(20,23)(24,35)(25,33)(26,30)(32,41)(42,44);(13,17,18,34,19,15,27,28)(14,23,24,33,22,20,35,25)(16,26,32,42,31,30,41,44)(21,36,38,39,29,37,43,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=f,c^2=h,d^2=gi,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511570);