Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1219 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)(25,26,31,32,30,35,29,36)(27,37,43,48,39,33,40,45)(28,34,44,46,42,49,41,50)(38,51,54,56,53,47,52,55);(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)(28,41)(29,31)(33,45)(35,36)(37,48)(38,52)(40,43)(42,44)(46,50)(53,54)(55,56);(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)(28,38)(29,40)(30,39)(31,43)(32,45)(34,47)(35,37)(36,48)(41,52)(42,53)(44,54)(46,55)(49,51)(50,56);(1,3,5,4)(2,6,8,7)(12,20)(15,24)(17,19)(21,23)(25,28,29,41,30,42,31,44)(26,34,36,50,35,49,32,46)(27,38,40,52,39,53,43,54)(33,47,48,56,37,51,45,55);(1,4,5,3)(2,7,8,6)(10,16)(11,18)(15,24)(17,19)(26,35)(27,39)(32,36)(34,49)(38,53)(40,43)(46,50)(52,54)Polycyclic presentation:
2^9|a^2=fgh,b^2=gi,c^2=ghi,d^2=fgi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7030990);