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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,5,7,8)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,31,32,30,36,29,35)(27,37,44,49,39,33,40,46)(28,34,45,47,43,52,41,51)(38,50,54,56,42,48,53,55);(1,3,4,7)(2,5,6,8)(11,18)(12,21)(14,17)(15,20)(26,32)(28,41)(29,31)(33,46)(35,36)(37,49)(38,53)(40,44)(42,54)(43,45)(47,51)(55,56);(2,6)(5,8)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,27)(26,33)(28,42)(29,40)(30,39)(31,44)(32,46)(34,50)(35,49)(36,37)(38,43)(41,54)(45,53)(47,56)(48,52)(51,55);(11,18)(14,17)(19,24)(22,23)(25,28,31,45,30,43,29,41)(26,34,32,47,36,52,35,51)(27,38,44,54,39,42,40,53)(33,48,46,55,37,50,49,56);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(27,39)(33,37)(38,42)(40,44)(46,49)(48,50)(53,54)(55,56)Polycyclic presentation:
2^9|a^2=fg,b^2=ghi,d^2=fi,e^2=h,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=dhi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7092211);