Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632088 |
| 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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,29,34)(27,36,37,31)(28,40,42,33)(30,32,44,35)(38,46,41,49)(39,45,51,47)(43,50,54,48)(52,56,53,55);(1,2)(3,6)(4,5)(7,8)(9,11,13,17)(10,14,16,21)(12,19,20,18)(15,23,24,22)(25,27)(26,31)(28,41)(29,37)(30,39)(32,47)(33,49)(34,36)(35,45)(38,42)(40,46)(43,53)(44,51)(48,55)(50,56)(52,54);(11,17)(14,21)(18,19)(22,23)(26,32)(27,37)(30,44)(31,45)(33,48)(34,35)(36,47)(38,41)(40,50)(43,54)(46,55)(49,56);(2,5)(6,7)(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,28,29,42)(26,33,34,40)(27,38,37,41)(30,43,44,54)(31,46,36,49)(32,48,35,50)(39,52,51,53)(45,55,47,56);(1,3,4,8)(2,6,5,7)(10,16)(14,21)(15,24)(22,23)(27,37)(28,42)(31,36)(33,40)(39,51)(43,54)(45,47)(48,50)Polycyclic presentation:
2^9|a^2=f,b^2=hi,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,e^b=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6556185);