Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907909 |
| 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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,19)(12,13)(14,17)(15,18)(16,22)(20,23)(21,24)(25,26)(27,35)(28,39)(29,33)(30,43)(31,37)(32,41)(34,42)(36,49)(38,45)(40,54)(44,56)(46,55)(47,51)(48,52)(50,53);(2,5)(3,7)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,24)(18,19)(25,27)(26,31)(28,36)(29,37)(30,38)(32,44)(33,35)(34,45)(39,53)(40,50)(41,51)(42,52)(43,48)(46,47)(49,54)(55,56);(1,2)(3,6)(4,5)(7,8)(9,12,15,17)(10,13,18,14)(11,20,22,24)(16,21,19,23)(25,28,30,41,29,40,42,55)(26,32,34,39,33,46,43,54)(27,36,38,51,37,50,52,56)(31,44,45,53,35,47,48,49);(2,5)(6,8)(9,13,15,14)(10,17,18,12)(11,21,22,23)(16,24,19,20)(26,33)(27,37)(32,46)(34,43)(36,50)(38,52)(39,54)(51,56);(1,3,4,7)(2,6,5,8)(9,14,15,13)(10,17,18,12)(11,21,22,23)(16,20,19,24)(27,37)(31,35)(36,50)(38,52)(44,47)(45,48)(49,53)(51,56)Polycyclic presentation:
2^9|c^2=fgi,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6878944);