Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1602 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)(7,11)(10,12)(13,14,18,21)(15,16,23,19)(17,20,25,27)(22,24,28,26)(29,30)(31,39)(32,36)(33,34)(35,38)(37,44)(40,41)(42,43);(6,9)(10,12)(17,25)(20,27)(22,28)(24,26)(29,31,35,32)(30,36,38,39)(33,42,44,40)(34,41,37,43);(5,6)(7,10)(8,9)(11,12)(13,15,18,23)(14,19,21,16)(17,22,25,28)(20,26,27,24)(29,32,35,31)(30,36,38,39)(33,40,44,42)(34,41,37,43);(1,2,3,4)(13,16,18,19)(14,15,21,23)(17,24,25,26)(20,22,27,28)(29,33)(30,37)(31,40)(32,42)(34,38)(35,44)(36,41)(39,43);(5,7)(6,10)(8,11)(9,12)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)(29,34,35,37)(30,33,38,44)(31,41,32,43)(36,42,39,40)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fg,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 4987337);