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)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,34)(35,40)(36,42)(37,39)(38,41)(43,47)(44,48)(45,46);(2,5)(6,8)(10,13)(14,16)(18,19)(21,29)(23,25)(27,32)(33,35,39,36)(34,40,37,42)(38,44,46,43)(41,48,45,47);(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)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(9,11)(10,14)(12,15)(13,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,37,39,34)(35,42,36,40)(38,45,46,41)(43,48,44,47);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=hi,d^2=fg,e^2=f,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, 5400188);