Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1347 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10,15,12)(11,18,20,16)(13,21,14,17)(19,23,22,24)(25,26)(27,36)(28,40)(29,35)(30,32)(31,44)(33,39)(34,42)(37,48)(38,53)(41,50)(43,49)(45,46)(47,54)(51,52)(55,56);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,18)(13,19)(14,22)(15,20)(17,23)(21,24)(26,32)(27,37)(34,40)(35,44)(38,52)(39,45)(49,50)(54,55);(3,8)(5,7)(9,12,15,10)(11,18,20,16)(13,17,14,21)(19,23,22,24)(25,27)(26,33)(28,38)(29,39)(30,37)(31,45)(32,46)(34,47)(35,48)(36,44)(40,56)(41,54)(42,52)(43,55)(49,51)(50,53);(2,6)(5,7)(9,13,15,14)(10,17,12,21)(11,19,20,22)(16,23,18,24)(25,28,30,42)(26,34,32,40)(27,38,37,52)(29,43,31,41)(33,47,46,56)(35,50,44,49)(36,51,48,53)(39,55,45,54);(9,14,15,13)(10,17,12,21)(11,19,20,22)(16,24,18,23)(25,29,30,31)(26,35,32,44)(27,39,37,45)(28,41,42,43)(33,48,46,36)(34,49,40,50)(38,54,52,55)(47,51,56,53)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=fh,d^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6901034);