Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324820 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26,30,34)(27,35,36,48)(28,32,40,46)(29,42,37,33)(31,43,44,39)(38,49,50,56)(41,53,51,47)(45,54,55,52);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(27,36)(28,40)(31,44)(32,46)(35,48)(39,43)(41,51)(47,53);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,37,43,30,36,29,39)(26,31,42,48,34,44,33,35)(28,38,51,54,40,50,41,52)(32,45,53,56,46,55,47,49);(2,6)(3,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,32)(27,38)(29,41)(30,40)(31,45)(33,47)(34,46)(35,49)(36,50)(37,51)(39,52)(42,53)(43,54)(44,55)(48,56);(2,6)(5,7)(10,18)(15,19)(16,17)(23,24)(28,40)(32,46)(38,50)(41,51)(45,55)(47,53)(49,56)(52,54)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fgi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dhi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7238724);