Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1335 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,34,31,35,44,50)(27,36,38,46,39,49,29,32)(28,40,43,56,42,33,47,48)(37,52,54,45,53,51,41,55);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,39)(26,32,35,46)(28,37,42,53)(29,44,38,30)(33,45,40,55)(34,49,50,36)(41,47,54,43)(48,52,56,51);(2,6)(5,7)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28)(26,33)(27,37)(29,41)(30,43)(31,42)(32,45)(34,48)(35,40)(36,51)(38,54)(39,53)(44,47)(46,55)(49,52)(50,56);(3,8)(5,7)(12,20)(15,24)(17,19)(21,23)(25,29,30,27,31,38,44,39)(26,32,34,36,35,46,50,49)(28,41,43,37,42,54,47,53)(33,45,48,51,40,55,56,52);(10,16)(11,18)(15,24)(17,19)(28,42)(33,40)(37,53)(41,54)(43,47)(45,55)(48,56)(51,52)Polycyclic presentation:
2^9|a^2=fghi,b^2=i,c^2=h,d^2=fgi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7018073);