Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907869 |
| 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 | |
1357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,15,17)(11,18,21,16)(12,13,22,14)(19,23,24,20)(25,26)(27,32)(28,40)(29,34)(30,44)(31,35)(33,42)(36,49)(37,46)(38,47)(39,53)(41,48)(43,54)(45,52)(50,56)(51,55);(1,2)(3,7)(4,6)(5,8)(9,11,15,21)(10,16,17,18)(12,19,22,24)(13,20,14,23)(25,27)(26,32)(28,36)(29,38)(30,39)(31,37)(33,45)(34,47)(35,46)(40,49)(41,51)(42,52)(43,50)(44,53)(48,55)(54,56);(1,3)(2,5)(4,8)(6,7)(9,12)(10,13)(11,19)(14,17)(15,22)(16,20)(18,23)(21,24)(25,28,29,41,31,43,30,42)(26,33,44,54,35,48,34,40)(27,36,38,51,37,50,39,52)(32,45,53,56,46,55,47,49);(3,8)(5,7)(9,13,15,14)(10,12,17,22)(11,20,21,23)(16,19,18,24)(27,37)(32,46)(36,50)(38,39)(45,55)(47,53)(49,56)(51,52);(2,6)(5,7)(9,14,15,13)(10,12,17,22)(11,20,21,23)(16,24,18,19)(25,29,31,30)(26,34,35,44)(27,38,37,39)(28,41,43,42)(32,47,46,53)(33,40,48,54)(36,51,50,52)(45,49,55,56)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gi,d^2=f,e^2=fi,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=dfh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6821994);