Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,15,16,13)(18,22)(19,25)(24,29)(27,31)(34,36)(35,41)(37,45)(46,48);(2,4)(3,8)(11,16)(13,15)(18,22)(23,26)(24,29)(30,32)(33,34,38,36)(35,39,41,42)(37,44,45,40)(43,48,47,46);(3,8)(6,7)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,3,5,8)(2,6,4,7)(9,11)(10,13)(12,16)(14,15)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,4,5,2)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=fhi,b^2=gh,c^2=h,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5908904);