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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1439 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)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26)(27,33)(28,35)(29,44)(30,36)(31,46)(32,37)(34,43)(38,48)(39,53)(40,49)(41,55)(42,50)(45,51)(47,52)(54,56);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,32,42)(26,33,37,50)(28,38,29,39)(30,40,31,41)(34,47,51,56)(35,48,44,53)(36,49,46,55)(43,52,45,54);(3,7)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,32,29)(26,34,37,51)(27,38,42,39)(30,43,31,45)(33,47,50,56)(35,36,44,46)(40,52,41,54)(48,49,53,55);(2,6)(3,7)(11,19)(14,17)(18,24)(21,23)(25,29,32,28)(26,35,37,44)(27,39,42,38)(30,43,31,45)(33,48,50,53)(34,36,51,46)(40,52,41,54)(47,49,56,55);(3,7)(5,8)(10,16)(14,17)(15,22)(21,23)(25,30,32,31)(26,36,37,46)(27,40,42,41)(28,43,29,45)(33,49,50,55)(34,44,51,35)(38,52,39,54)(47,53,56,48)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dhi,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, 7108529);