Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,32)(28,34)(29,36)(30,35)(31,37)(33,42)(38,54)(39,47)(40,53)(41,49)(43,52)(44,48)(45,51)(46,56)(50,55);(1,3,5,8)(2,6,4,7)(11,17)(12,20)(14,21)(15,19)(25,27,31,41)(26,32,37,49)(28,39,43,38)(29,44,30,40)(33,50,51,46)(34,47,52,54)(35,53,36,48)(42,55,45,56);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,28,31,43)(26,33,37,51)(27,38,41,39)(29,42,30,45)(32,46,49,50)(34,36,52,35)(40,55,44,56)(47,53,54,48);(1,4,5,2)(3,6,8,7)(11,17)(14,21)(18,23)(22,24)(25,28)(26,34)(27,39)(29,45)(30,42)(31,43)(32,47)(33,35)(36,51)(37,52)(38,41)(40,55)(44,56)(46,48)(49,54)(50,53);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,29,31,30)(26,35,37,36)(27,40,41,44)(28,42,43,45)(32,48,49,53)(33,34,51,52)(38,55,39,56)(46,47,50,54)Polycyclic presentation:
2^9|a^2=fhi,b^2=fi,c^2=i,d^2=f,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,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7075983);