Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1333 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)(27,36)(28,33)(29,34)(30,32)(31,38)(35,42)(37,49)(39,45)(40,47)(41,46)(43,52)(44,50)(48,53)(51,55)(54,56);(1,2)(3,7)(4,6)(5,8)(9,11,13,18)(10,14,16,17)(12,19,21,24)(15,22,20,23)(25,27,29,38)(26,31,34,36)(28,37,40,50)(30,39,42,52)(32,43,35,45)(33,44,47,49)(41,51,53,56)(46,54,48,55);(1,3)(2,5)(4,8)(6,7)(26,32)(30,42)(31,43)(33,46)(34,35)(36,45)(39,52)(41,53)(44,54)(47,48)(49,55)(51,56);(2,6)(3,8)(11,18)(14,17)(19,24)(22,23)(25,28,29,40)(26,33,34,47)(27,37,38,50)(30,41,42,53)(31,44,36,49)(32,46,35,48)(39,51,52,56)(43,54,45,55);(2,6)(5,7)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(26,34)(28,40)(31,36)(32,35)(37,50)(41,53)(43,45)(51,56)Polycyclic presentation:
2^9|a^2=fi,b^2=f,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=dfhi,e^b=eh,d^c=dh,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, 7213923);