Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1435 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(11,15)(14,16)(17,18)(19,20)(21,26)(22,23)(24,25)(27,28)(29,32)(30,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(10,13)(14,16)(18,23)(19,25)(26,31)(27,29)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,23)(21,29)(22,25)(26,28)(27,30)(31,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(1,2)(3,6)(4,5)(7,8)(17,20,22,24)(18,25,23,19)(21,28,30,32)(26,29,31,27)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,21,22,30)(18,26,23,31)(19,27,25,29)(20,28,24,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)Polycyclic presentation:
2^9|b^2=fg,c^2=fh,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5468465);