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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1361 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)(28,38,36,33,41,46,40,49)(29,42,45,47,43,34,37,51)(39,53,50,56,54,52,55,48);(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)(29,37,43,45)(33,46,49,38)(34,47,42,51)(39,50,54,55)(48,53,56,52);(1,2)(3,8)(4,7)(5,6)(25,28,30,41)(26,33,35,49)(27,36,31,40)(29,39,43,54)(32,46,44,38)(34,48,42,56)(37,50,45,55)(47,53,51,52);(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)(31,45)(32,47)(33,48)(35,42)(36,50)(38,52)(40,55)(41,54)(44,51)(46,53)(49,56);(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)(34,42)(37,45)(39,54)(47,51)(48,56)(50,55)(52,53)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,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, 7198572);