Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500434 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1386 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)(4,5)(6,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,36)(28,41)(29,34)(30,45)(31,35)(32,39)(33,42)(37,52)(38,47)(40,51)(43,49)(44,48)(46,55)(50,53)(54,56);(2,5)(6,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,38,31,40,30,39)(26,32,45,51,35,47,34,36)(28,37,44,55,42,53,43,54)(33,46,48,52,41,56,49,50);(1,3,4,8)(2,6,5,7)(11,17)(12,19)(14,20)(15,22)(25,28,31,42)(26,33,35,41)(27,37,40,53)(29,44,30,43)(32,46,47,56)(34,49,45,48)(36,50,51,52)(38,55,39,54);(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(28,42)(33,41)(37,53)(43,44)(46,56)(48,49)(50,52)(54,55);(11,17)(14,20)(18,23)(21,24)(25,29,31,30)(26,34,35,45)(27,38,40,39)(28,43,42,44)(32,36,47,51)(33,48,41,49)(37,54,53,55)(46,52,56,50)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6435746);