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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32)(34,38)(35,41)(40,44)(43,45);(11,15)(13,16)(18,22)(20,29)(23,25)(26,30)(33,34,37,38)(35,42,41,39)(36,44,46,40)(43,47,45,48);(1,2,4,6)(3,5,7,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,29)(30,31)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,46)(45,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,27)(18,23,22,25)(20,30,29,26)(24,32,28,31)(33,35)(34,39)(36,45)(37,41)(38,42)(40,48)(43,46)(44,47);(17,20,21,29)(18,24,22,28)(19,26,27,30)(23,31,25,32)(33,36,37,46)(34,40,38,44)(35,43,41,45)(39,47,42,48)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=hi,d^2=fg,e^2=f,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, 3014978);