Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1347 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,29,34)(27,36,38,31)(28,33,40,48)(30,32,42,35)(37,49,50,44)(39,45,52,43)(41,46,53,47)(51,55,56,54);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,29,38)(26,31,34,36)(28,37,40,50)(30,39,42,52)(32,43,35,45)(33,44,48,49)(41,51,53,56)(46,54,47,55);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(26,32)(28,40)(30,42)(31,43)(33,47)(34,35)(36,45)(37,50)(39,52)(44,55)(46,48)(49,54);(11,18)(14,22)(17,19)(21,23)(25,28,29,40)(26,33,34,48)(27,37,38,50)(30,41,42,53)(31,44,36,49)(32,46,35,47)(39,51,52,56)(43,54,45,55);(1,3,4,7)(2,6,5,8)(10,16)(14,22)(15,24)(21,23)(27,38)(28,40)(31,36)(33,48)(39,52)(41,53)(43,45)(46,47)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6386634);