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 | |
1740 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)(7,11)(8,10)(9,12)(13,14,18,15)(16,21,23,19)(17,20,25,22)(24,27,28,26)(29,30)(31,36)(32,33)(34,37)(35,38)(39,43)(40,41)(42,44);(5,6)(7,11)(8,10)(9,12)(14,15)(16,23)(20,22)(24,28)(29,31,35,33)(30,36,38,32)(34,41,42,39)(37,40,44,43);(5,6,8,10)(7,9,12,11)(13,15,18,14)(16,19,23,21)(17,20,25,22)(24,27,28,26)(29,32)(30,31)(33,38)(34,40)(35,36)(37,39)(41,44)(42,43);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27)(29,33,35,31)(30,36,38,32)(34,41,42,39)(37,43,44,40);(5,7,8,12)(6,9,10,11)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)(29,34,35,42)(30,37,38,44)(31,39,33,41)(32,40,36,43)Polycyclic presentation:
2^9|a^2=f,b^2=fgh,c^2=h,d^2=i,e^2=g,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, 5187998);