Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1171 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26)(27,39)(28,43)(29,35)(30,38)(31,37)(32,36)(33,41)(34,42)(40,53)(44,52)(45,51)(46,50)(47,56)(48,55)(49,54);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,31,28,32,42,30,41)(26,33,38,34,36,43,37,39)(29,45,49,46,47,40,48,44)(35,52,55,53,56,50,54,51);(1,3,5,4)(2,7,8,6)(11,17)(14,21)(18,19)(22,23)(25,28,31,42,32,41,30,27)(26,34,38,43,36,39,37,33)(29,46,49,40,47,44,48,45)(35,53,55,50,56,51,54,52);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29,32,47)(26,35,36,56)(27,40,42,45)(28,44,41,46)(30,48,31,49)(33,50,43,52)(34,51,39,53)(37,54,38,55);(1,4,5,3)(2,6,8,7)(10,16)(14,21)(15,24)(22,23)(26,36)(29,47)(33,43)(34,39)(37,38)(40,45)(44,46)(48,49)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fi,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6393763);