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)(27,33)(28,41)(29,35)(30,36)(31,46)(32,37)(34,43)(38,48)(39,49)(40,51)(42,56)(44,53)(45,50)(47,55)(52,54);(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)(26,33)(28,42)(29,38)(30,45)(31,39)(32,40)(34,52)(35,48)(36,50)(37,51)(41,56)(43,54)(44,55)(46,49)(47,53);(3,8)(5,7)(9,12,15,10)(11,18,20,16)(13,17,14,21)(19,23,22,24)(25,28)(26,34)(27,38)(29,32)(30,43)(31,44)(33,47)(35,36)(37,53)(39,54)(40,42)(41,46)(45,55)(48,49)(50,56)(51,52);(2,6)(5,7)(9,13,15,14)(10,17,12,21)(11,19,20,22)(16,23,18,24)(25,29,32,28)(26,35,37,41)(27,38,40,42)(30,43,31,44)(33,48,51,56)(34,46,53,36)(39,55,45,54)(47,50,52,49);(9,14,15,13)(10,17,12,21)(11,19,20,22)(16,24,18,23)(25,30,32,31)(26,36,37,46)(27,39,40,45)(28,43,29,44)(33,49,51,50)(34,35,53,41)(38,54,42,55)(47,48,52,56)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=fhi,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, 6901019);