Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1255 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)(3,6)(4,5)(7,8)(11,17)(14,18)(19,20)(22,23)(25,26,31,33,29,35,30,36)(27,37,44,46,38,32,40,48)(28,41,45,50,42,34,43,49)(39,47,54,56,52,51,53,55);(1,3,7,4)(2,5,8,6)(9,10)(11,18)(12,15)(13,16)(14,17)(19,23)(20,22)(21,24)(25,27)(26,32)(28,39)(29,38)(30,40)(31,44)(33,48)(34,47)(35,37)(36,46)(41,51)(42,52)(43,53)(45,54)(49,56)(50,55);(1,4,7,3)(2,5,8,6)(9,11,13,17)(10,14,16,18)(12,20,21,19)(15,23,24,22)(26,33)(27,38)(30,31)(32,46)(34,49)(35,36)(37,48)(39,52)(41,50)(43,45)(47,55)(51,56);(1,5,7,6)(2,3,8,4)(9,12,13,21)(10,15,16,24)(11,19,17,20)(14,22,18,23)(25,28,29,42)(26,34,35,41)(27,39,38,52)(30,43,31,45)(32,47,37,51)(33,49,36,50)(40,53,44,54)(46,55,48,56);(1,6,7,5)(2,3,8,4)(10,16)(14,18)(15,24)(22,23)(28,42)(34,41)(39,52)(43,45)(47,51)(49,50)(53,54)(55,56)Polycyclic presentation:
2^9|a^2=gh,b^2=fi,c^2=fh,d^2=fhi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,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, 6870416);