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 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1206 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,17,19,14)(12,16,21,23)(18,24,20,22)(25,26,29,33)(27,35,38,31)(28,32,40,48)(30,34,42,50)(36,53,54,45)(37,51,43,46)(39,56,55,47)(41,49,44,52);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(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);(2,5)(6,8)(25,28)(26,32)(27,36)(29,40)(30,44)(31,45)(33,48)(34,52)(35,53)(37,55)(38,54)(39,43)(41,42)(46,56)(47,51)(49,50);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(26,33)(28,40)(31,35)(34,50)(36,54)(39,55)(41,44)(46,51);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(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=f,b^2=fi,d^2=g,e^2=fi,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, 6611945);