Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1437 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)(17,18,21,24)(19,25,26,22)(20,28,29,23)(27,30,32,31)(33,34,38,35)(36,41,44,39)(37,42,45,40)(43,47,46,48);(9,10,13,12)(11,15,16,14)(19,26)(22,25)(27,32)(30,31)(34,35)(37,45)(39,41)(43,46);(1,2,5,4)(3,6,7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,35,38,34)(36,39,44,41)(37,42,45,40)(43,47,46,48);(1,3)(2,6)(4,8)(5,7)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,31)(29,32)(33,36,38,44)(34,39,35,41)(37,46,45,43)(40,48,42,47);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,15,16,14)(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)(33,37,38,45)(34,40,35,42)(36,43,44,46)(39,47,41,48)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=fghi,d^2=f,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3164328);