Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,6)(3,7,4,8)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,27,35,30,34,38,31)(28,39,42,48,41,50,36,32)(29,43,37,46,45,33,44,49)(40,53,54,47,55,52,51,56);(2,6)(3,4)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,27,30,38)(26,31,34,35)(28,36,41,42)(29,44,45,37)(32,39,48,50)(33,49,43,46)(40,54,55,51)(47,53,56,52);(1,2)(3,8)(4,7)(5,6)(25,28,30,41)(26,32,34,48)(27,36,38,42)(29,40,45,55)(31,39,35,50)(33,47,43,56)(37,51,44,54)(46,52,49,53);(1,3,5,4)(2,7,6,8)(11,18)(14,17)(19,24)(22,23)(25,29)(26,33)(27,37)(28,40)(30,45)(31,46)(32,47)(34,43)(35,49)(36,51)(38,44)(39,52)(41,55)(42,54)(48,56)(50,53);(1,4,5,3)(2,7,6,8)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(26,34)(29,45)(31,35)(32,48)(37,44)(39,50)(40,55)(51,54)Polycyclic presentation:
2^9|a^2=fghi,b^2=i,c^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7198573);