Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(9,12)(13,14,17,18)(15,19,22,26)(16,23,25,20)(21,28,24,27)(29,30)(31,36)(32,37)(33,38)(34,35)(39,42)(40,41)(43,44);(2,4)(5,6,8,10)(7,9,11,12)(14,18)(19,26)(20,23)(27,28)(30,35)(32,36)(38,44)(40,42);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,23)(17,18)(20,25)(21,28)(22,26)(24,27)(29,31,34,37)(30,36,35,32)(33,41,43,39)(38,40,44,42);(5,7)(6,9)(8,11)(10,12)(13,15,17,22)(14,19,18,26)(16,24,25,21)(20,28,23,27)(29,32,34,36)(30,37,35,31)(33,42,43,40)(38,39,44,41);(13,16,17,25)(14,20,18,23)(15,21,22,24)(19,27,26,28)(29,33,34,43)(30,38,35,44)(31,39,37,41)(32,40,36,42)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3272869);