Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,6,4,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26)(27,35)(28,32)(29,33)(30,34)(31,38)(36,53)(37,51)(39,56)(40,48)(41,49)(42,50)(43,46)(44,52)(45,54)(47,55);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27)(26,31)(28,39)(29,38)(30,43)(32,47)(33,35)(34,51)(36,41)(37,42)(40,55)(44,54)(45,49)(46,50)(48,56)(52,53);(2,7)(6,8)(25,28,29,40)(26,32,33,48)(27,36,38,54)(30,44,42,41)(31,45,35,53)(34,52,50,49)(37,55,43,39)(46,56,51,47);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(26,33)(28,40)(31,35)(34,50)(36,54)(39,55)(41,44)(46,51);(1,4,5,3)(2,8,7,6)(11,19)(14,17)(18,24)(22,23)(27,37)(30,42)(31,46)(34,50)(35,51)(36,55)(38,43)(39,54)(41,44)(45,56)(47,53)(49,52)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=i,d^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6626670);