Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1345 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26)(27,36)(28,41)(29,34)(30,33)(31,38)(32,44)(35,42)(37,51)(39,49)(40,46)(43,47)(45,52)(48,56)(50,54)(53,55);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27)(26,31)(28,37)(29,40)(30,38)(32,45)(33,36)(34,49)(35,46)(39,42)(41,55)(43,53)(44,54)(47,51)(48,50)(52,56);(2,6)(3,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,42,56,30,43,29,44)(26,32,34,47,33,48,35,41)(27,37,39,52,38,53,40,54)(31,45,49,51,36,50,46,55);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,33)(27,38)(28,43)(34,35)(39,40)(44,56)(45,50)(51,55);(3,7)(5,8)(11,21)(15,18)(19,20)(23,24)(25,29,30,42)(26,34,33,35)(27,39,38,40)(28,44,43,56)(31,46,36,49)(32,47,48,41)(37,52,53,54)(45,55,50,51)Polycyclic presentation:
2^9|a^2=fi,c^2=fgh,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6670919);