Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1347 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10,13,16)(11,17,18,14)(12,15,20,22)(19,23,24,21)(25,26)(27,37)(28,42)(29,34)(30,35)(31,36)(32,40)(33,44)(38,52)(39,48)(41,47)(43,50)(45,49)(46,53)(51,54)(55,56);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,17)(20,24)(22,23)(25,27)(26,32)(28,38)(29,40)(30,41)(31,39)(33,46)(34,48)(35,37)(36,47)(42,56)(43,54)(44,55)(45,53)(49,51)(50,52);(11,18)(14,17)(19,24)(21,23)(25,28,31,45)(26,33,36,50)(27,38,39,53)(29,43,30,44)(32,46,47,52)(34,42,35,49)(37,51,48,56)(40,54,41,55);(1,3,4,7)(2,6,5,8)(12,20)(15,22)(19,24)(21,23)(27,39)(32,47)(37,48)(38,53)(40,41)(46,52)(51,56)(54,55);(9,12,13,20)(10,15,16,22)(11,19,18,24)(14,21,17,23)(25,29,31,30)(26,34,36,35)(27,40,39,41)(28,43,45,44)(32,48,47,37)(33,42,50,49)(38,54,53,55)(46,56,52,51)Polycyclic presentation:
2^9|a^2=fg,c^2=i,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6451806);