Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533141 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #3459 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(17,18)(19,29)(20,34,37,25)(21,27)(22,26)(23,28)(24,33)(30,43,46,38)(31,39)(32,40)(35,41,36,42)(44,47,45,48);(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,38)(26,39)(27,40)(28,29)(34,43)(35,44)(36,45)(37,46)(41,47)(42,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,41)(27,42)(28,34)(29,43)(31,44)(32,45)(33,46)(39,47)(40,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,39,29,40)(25,41,34,42)(30,44,46,45)(38,47,43,48);(4,12)(7,14)(10,16)(13,15)(17,22,23,21)(18,27,28,26)(19,32,33,31)(20,36,37,35)(24,40,29,39)(25,42,34,41)(30,45,46,44)(38,48,43,47)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=h,d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,g^b=gi,f^c=fi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6479774);