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 | |
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)(13,14,18,21)(15,16,23,19)(17,20,25,27)(22,24,28,26)(29,30,35,37)(31,38,32,36)(33,43,44,34)(39,42,41,40);(5,6)(7,11)(8,10)(9,12)(17,25)(20,27)(22,28)(24,26)(29,31,35,32)(30,36,37,38)(33,41,44,39)(34,40,43,42);(5,6,8,10)(7,9,12,11)(13,15,18,23)(14,19,21,16)(17,22,25,28)(20,26,27,24)(29,32,35,31)(30,36,37,38)(33,39,44,41)(34,40,43,42);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,19)(14,15,21,23)(17,24,25,26)(20,22,27,28)(29,33)(30,34)(31,39)(32,41)(35,44)(36,42)(37,43)(38,40);(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)(30,33)(31,40)(32,42)(35,43)(36,41)(37,44)(38,39)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=fg,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, 4990191);