Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632048 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1271 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)(25,26,29,33,30,34,44,50)(27,35,39,46,38,31,37,47)(28,40,43,49,42,48,41,32)(36,52,53,45,54,56,55,51);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,38)(26,31,34,35)(28,36,42,54)(29,39,44,37)(32,45,49,51)(33,47,50,46)(40,56,48,52)(41,55,43,53);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28)(26,32)(27,36)(29,41)(30,42)(31,45)(33,48)(34,49)(35,51)(37,53)(38,54)(39,55)(40,50)(43,44)(46,56)(47,52);(2,5)(6,7)(11,18)(14,22)(17,19)(21,23)(25,29,30,44)(26,33,34,50)(27,37,38,39)(28,41,42,43)(31,46,35,47)(32,48,49,40)(36,53,54,55)(45,56,51,52);(1,3,4,8)(2,6,5,7)(10,16)(14,22)(15,24)(21,23)(27,38)(31,35)(36,54)(37,39)(45,51)(46,47)(52,56)(53,55)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=h,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=efhi,c^b=ch,d^b=dfh,e^b=ei,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6551643);