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 | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1393 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)(25,26,30,32)(27,37,40,50)(28,34,41,46)(29,36,31,35)(33,44,45,39)(38,51,52,56)(42,49,43,48)(47,53,55,54);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,15,17,18)(12,19,13,20)(16,23,22,24)(26,32)(28,41)(33,45)(35,36)(37,50)(38,52)(42,43)(53,54);(3,7)(5,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27)(26,33)(28,38)(29,39)(30,40)(31,44)(32,45)(34,47)(35,37)(36,50)(41,52)(42,53)(43,54)(46,55)(48,56)(49,51);(2,6)(3,7)(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,24,18,23)(25,28,30,41)(26,34,32,46)(27,38,40,52)(29,43,31,42)(33,47,45,55)(35,49,36,48)(37,51,50,56)(39,54,44,53);(3,7)(5,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29,30,31)(26,35,32,36)(27,39,40,44)(28,42,41,43)(33,37,45,50)(34,48,46,49)(38,53,52,54)(47,56,55,51)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=fi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,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, 7114902);