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 | |
1502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(11,15)(14,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(2,5)(6,8)(10,13)(14,16)(18,19)(21,29)(23,25)(27,32)(33,35,38,39)(34,36,41,42)(37,43,46,45)(40,44,48,47);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,25,28,23)(21,24,29,26)(27,30,32,31)(33,35)(34,36)(37,45)(38,39)(40,47)(41,42)(43,46)(44,48);(9,11)(10,14)(12,15)(13,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36)(34,39)(35,41)(37,44)(38,42)(40,45)(43,48)(46,47);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,46)(34,40,41,48)(35,43,39,45)(36,44,42,47)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fghi,d^2=fg,e^2=g,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, 5400816);