Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10,14,15)(11,16,13,12)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32)(33,34,38,36)(35,39,41,42)(37,44,45,40)(43,48,47,46);(9,11,14,13)(10,12,15,16)(18,22)(19,26)(24,28)(27,32)(35,41)(39,42)(43,47)(46,48);(9,12)(10,11)(13,15)(14,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,2)(3,6)(4,5)(7,8)(9,13,14,11)(10,12,15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,3,4,7)(2,6,5,8)(9,12)(10,13)(11,15)(14,16)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fh,d^2=gh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4819482);