Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,19)(16,18)(17,20)(21,23)(22,27)(24,26)(25,28)(29,30,34,35)(31,32,39,36)(33,41,42,37)(38,44,43,40);(5,6,9,12)(7,8,11,10)(14,16)(15,21)(20,24)(22,25)(30,35)(32,36)(33,42)(38,43);(7,11)(8,10)(13,15,18,21)(14,19,16,23)(17,25,26,22)(20,28,24,27)(29,31)(30,36)(32,35)(33,43)(34,39)(37,40)(38,42)(41,44);(1,2,3,4)(5,7,9,11)(6,10,12,8)(13,16,18,14)(15,19,21,23)(17,24,26,20)(22,27,25,28)(29,32,34,36)(30,31,35,39)(33,40,42,44)(37,38,41,43);(5,8,9,10)(6,11,12,7)(13,17,18,26)(14,20,16,24)(15,22,21,25)(19,27,23,28)(29,33,34,42)(30,37,35,41)(31,38,39,43)(32,40,36,44)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=h,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5187533);