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 | |
1600 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)(8,9)(10,12)(13,14,17,18)(15,21,22,19)(16,24,25,20)(23,27,26,28)(29,30)(31,37)(32,40)(33,34)(35,39)(36,41)(38,42)(43,44);(5,7,10,8)(6,11,12,9)(14,18)(15,22)(16,25)(27,28)(30,34)(32,41)(35,37)(38,44);(6,12)(7,8)(13,15,17,22)(14,19,18,21)(16,26,25,23)(20,28,24,27)(29,31,33,39)(30,35,34,37)(32,38,41,44)(36,42,40,43);(1,2,3,4)(5,8,10,7)(6,11,12,9)(13,14)(15,19)(16,20)(17,18)(21,22)(23,27)(24,25)(26,28)(29,32)(30,36)(31,38)(33,41)(34,40)(35,42)(37,43)(39,44);(5,9,10,11)(6,7,12,8)(13,16,17,25)(14,20,18,24)(15,23,22,26)(19,27,21,28)(29,30)(31,37)(32,36)(33,34)(35,39)(38,43)(40,41)(42,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=f,d^2=fi,e^2=h,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, 4990161);