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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1361 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)(12,15,21,24)(18,23,20,22)(25,26)(27,32)(28,34)(29,35)(30,44)(31,36)(33,40)(37,46)(38,47)(39,49)(41,50)(42,48)(43,51)(45,53)(52,55)(54,56);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,31,39)(26,32,36,49)(28,37,41,54)(29,42,30,38)(33,45,51,55)(34,46,50,56)(35,48,44,47)(40,53,43,52);(3,7)(5,8)(25,28)(26,33)(27,37)(29,40)(30,43)(31,41)(32,45)(34,44)(35,50)(36,51)(38,52)(39,54)(42,53)(46,47)(48,56)(49,55);(2,6)(3,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,34)(27,37)(29,43)(30,40)(31,41)(32,46)(33,44)(35,51)(36,50)(38,53)(39,54)(42,52)(45,47)(48,55)(49,56);(3,7)(5,8)(10,16)(14,17)(15,24)(22,23)(25,29,31,30)(26,35,36,44)(27,38,39,42)(28,40,41,43)(32,47,49,48)(33,50,51,34)(37,52,54,53)(45,46,55,56)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=f,e^2=i,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=ehi,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, 7110387);