Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234345 |
| 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)(25,26,31,44,30,35,27,32)(28,38,36,33,41,45,40,48)(29,34,42,49,43,50,37,46)(39,52,51,47,54,55,53,56);(2,6)(3,4)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,27,30,31)(26,32,35,44)(28,36,41,40)(29,42,43,37)(33,45,48,38)(34,49,50,46)(39,53,54,51)(47,52,56,55);(1,2)(3,8)(4,7)(5,6)(25,28,30,41)(26,33,35,48)(27,36,31,40)(29,39,43,54)(32,45,44,38)(34,47,50,56)(37,51,42,53)(46,55,49,52);(1,3,5,4)(2,7,6,8)(11,18)(14,17)(19,24)(22,23)(25,29)(26,34)(27,37)(28,39)(30,43)(31,42)(32,46)(33,47)(35,50)(36,51)(38,52)(40,53)(41,54)(44,49)(45,55)(48,56);(1,4,5,3)(2,7,6,8)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(29,43)(34,50)(37,42)(39,54)(46,49)(47,56)(51,53)(52,55)Polycyclic presentation:
2^9|a^2=fghi,b^2=i,c^2=i,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfi,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, 7198444);