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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1365 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)(25,26,29,34)(27,36,37,31)(28,40,41,33)(30,32,43,35)(38,45,50,49)(39,44,51,46)(42,48,53,47)(52,55,56,54);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,18)(13,19)(14,22)(15,20)(17,23)(21,24)(25,27)(26,31)(28,38)(29,37)(30,39)(32,46)(33,45)(34,36)(35,44)(40,49)(41,50)(42,52)(43,51)(47,55)(48,54)(53,56);(3,8)(5,7)(9,12,15,10)(11,18,20,16)(13,17,14,21)(19,23,22,24)(26,32)(27,37)(30,43)(31,44)(33,47)(34,35)(36,46)(38,50)(40,48)(42,53)(45,54)(49,55);(2,6)(5,7)(9,13,15,14)(10,17,12,21)(11,19,20,22)(16,23,18,24)(25,28)(26,33)(27,38)(29,41)(30,42)(31,45)(32,47)(34,40)(35,48)(36,49)(37,50)(39,52)(43,53)(44,54)(46,55)(51,56);(9,14,15,13)(10,17,12,21)(11,19,20,22)(16,24,18,23)(28,41)(33,40)(38,50)(42,53)(45,49)(47,48)(52,56)(54,55)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=fh,d^2=fh,e^2=fh,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, 6901015);