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) = C25×C4 |
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 | |
1367 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)(3,4)(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(35,41)(38,42)(43,47)(45,48);(1,2)(3,8)(4,7)(5,6)(11,16)(13,15)(18,22)(19,25)(20,28)(30,32)(33,34,37,40)(35,42,41,38)(36,39,44,46)(43,48,47,45);(2,6)(3,7)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,34)(35,38)(36,39)(37,40)(41,42)(43,45)(44,46)(47,48);(1,3,5,7)(2,4,6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35)(34,38)(36,43)(37,41)(39,45)(40,42)(44,47)(46,48);(1,4)(2,7)(3,6)(5,8)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,36,37,44)(34,39,40,46)(35,43,41,47)(38,45,42,48)Polycyclic presentation:
2^9|a^2=fghi,b^2=fgh,c^2=ghi,d^2=hi,e^2=gh,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, 6038690);