Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| 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 | |
1285 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)(3,6)(4,7)(5,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,36)(27,32,40,51)(28,41,43,34)(29,35,30,46)(33,45,53,42)(37,48,38,56)(39,49,44,50)(47,54,52,55);(3,4)(6,7)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27)(26,32)(28,38)(29,44)(30,39)(31,40)(33,52)(34,48)(35,50)(36,51)(37,43)(41,56)(42,54)(45,55)(46,49)(47,53);(1,3,5,4)(2,6,8,7)(25,28)(26,33)(27,37)(29,42)(30,45)(31,43)(32,47)(34,35)(36,53)(38,40)(39,54)(41,46)(44,55)(48,49)(50,56)(51,52);(1,3,5,4)(2,7,8,6)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,34)(27,38)(29,45)(30,42)(31,43)(32,48)(33,35)(36,41)(37,40)(39,54)(44,55)(46,53)(47,49)(50,52)(51,56);(1,4,5,3)(2,7,8,6)(10,16)(14,17)(15,24)(22,23)(25,29,31,30)(26,35,36,46)(27,39,40,44)(28,42,43,45)(32,49,51,50)(33,34,53,41)(37,54,38,55)(47,48,52,56)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=hi,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7141066);