Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(12,20)(15,21)(19,24)(22,23)(25,26)(27,32)(28,38)(29,34)(30,44)(31,35)(33,40)(36,47)(37,45)(39,50)(41,51)(42,48)(43,54)(46,53)(49,55)(52,56);(1,3)(2,6)(4,7)(5,8)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,27,31,30)(26,32,35,44)(28,36,41,40)(29,37,42,43)(33,38,47,51)(34,45,48,54)(39,49,52,53)(46,50,55,56);(1,2,5,4)(3,6,8,7)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,28,30,40,31,41,27,36)(26,33,32,38,35,47,44,51)(29,39,43,53,42,52,37,49)(34,46,45,50,48,55,54,56);(1,4,5,2)(3,6,8,7)(11,18)(14,17)(19,24)(22,23)(25,29,31,42)(26,34,35,48)(27,37,30,43)(28,39,41,52)(32,45,44,54)(33,46,47,55)(36,49,40,53)(38,50,51,56);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(29,42)(34,48)(37,43)(39,52)(45,54)(46,55)(49,53)(50,56)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=fg,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7094303);