Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,6,4,7)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,35,31,36,29,32)(27,33,39,47,37,49,43,45)(28,34,40,51,42,52,44,46)(38,48,54,56,41,50,53,55);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,32)(27,37)(28,40)(29,30)(33,47)(35,36)(38,53)(41,54)(42,44)(45,49)(46,51)(48,50);(1,3,5,4)(2,7,8,6)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,27)(26,33)(28,41)(29,43)(30,39)(31,37)(32,45)(34,50)(35,47)(36,49)(38,42)(40,53)(44,54)(46,56)(48,52)(51,55);(1,3,5,4)(2,6,8,7)(12,20)(15,24)(17,19)(21,23)(25,28,30,40,31,42,29,44)(26,34,35,51,36,52,32,46)(27,38,39,54,37,41,43,53)(33,48,47,56,49,50,45,55);(1,4,5,3)(2,7,8,6)(10,16)(11,18)(15,24)(17,19)(25,29,31,30)(26,35,36,32)(27,39,37,43)(28,40,42,44)(33,45,49,47)(34,46,52,51)(38,53,41,54)(48,56,50,55)Polycyclic presentation:
2^9|a^2=fghi,b^2=g,c^2=ghi,d^2=fg,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7030900);