Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1296 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)(9,10,14,12)(11,15,13,16)(17,18)(19,23)(20,22)(21,28)(24,30)(25,32)(26,27)(29,31)(33,34,37,38)(35,41,42,39)(36,40,44,46)(43,48,45,47);(9,11,14,13)(10,15,12,16)(18,20)(23,27)(24,28)(31,32)(34,38)(35,42)(36,44)(47,48);(9,12,14,10)(11,15,13,16)(17,19)(18,23)(20,27)(21,29)(22,26)(24,32)(25,30)(28,31)(33,35,37,42)(34,39,38,41)(36,45,44,43)(40,48,46,47);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,16,12,15)(17,20,22,18)(19,23,26,27)(21,28,30,24)(25,31,29,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,46)(45,48);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,15,12,16)(17,21)(18,24)(19,25)(20,28)(22,30)(23,31)(26,29)(27,32)(33,36,37,44)(34,40,38,46)(35,43,42,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=fh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5265843);