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, 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)(9,10,13,14)(11,12,16,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,32)(28,30)(29,31)(34,38)(35,41)(40,44)(43,47);(9,11)(10,12)(13,16)(14,15)(18,23)(19,25)(21,29)(28,32)(33,34,37,38)(35,39,41,42)(36,40,45,44)(43,46,47,48);(10,14)(11,16)(17,19,22,25)(18,24,23,20)(21,30,29,27)(26,28,31,32)(33,34)(35,42)(36,44)(37,38)(39,41)(40,45)(43,46)(47,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,20,22,24)(18,25,23,19)(21,28,29,32)(26,30,31,27)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,48)(45,47);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,11,14,16)(17,21)(18,26)(19,27)(20,28)(22,29)(23,31)(24,32)(25,30)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=fh,b^2=fgh,c^2=gh,d^2=gh,e^2=fghi,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, 5480118);