Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324500 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,30,35)(27,36,31,32)(28,39,41,45)(29,34,44,50)(33,37,48,42)(38,49,43,46)(40,52,54,55)(47,51,56,53);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,27,30,31)(26,32,35,36)(28,37,41,42)(29,43,44,38)(33,45,48,39)(34,49,50,46)(40,53,54,51)(47,52,56,55);(1,3)(2,5)(4,8)(6,7)(25,28,27,37,30,41,31,42)(26,33,32,45,35,48,36,39)(29,40,38,51,44,54,43,53)(34,47,46,55,50,56,49,52);(2,6)(3,8)(11,18)(14,17)(19,24)(22,23)(25,29,30,44)(26,34,35,50)(27,38,31,43)(28,40,41,54)(32,46,36,49)(33,47,48,56)(37,51,42,53)(39,52,45,55);(2,6)(5,7)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(26,35)(28,41)(29,44)(32,36)(37,42)(38,43)(47,56)(52,55)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7239423);