Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(11,15)(13,16)(17,18,22,19)(20,25,28,23)(21,24,30,26)(27,32,29,31)(33,34)(35,38)(36,41)(37,42)(39,44)(40,45)(43,47)(46,48);(2,5)(6,8)(9,10,12,14)(11,13,15,16)(18,19)(23,25)(24,26)(31,32)(34,35)(37,45)(39,41)(43,46);(1,2,4,5)(3,6,7,8)(17,19,22,18)(20,25,28,23)(21,24,30,26)(27,31,29,32)(33,35,38,34)(36,39,44,41)(37,40,45,42)(43,48,46,47);(9,11)(10,13)(12,15)(14,16)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,36,38,44)(34,39,35,41)(37,46,45,43)(40,48,42,47);(1,3,4,7)(2,6,5,8)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(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=fi,c^2=h,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4085636);