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 | |
1630 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)(7,9)(8,10)(11,12)(13,14,17,18)(15,21,23,19)(16,20,24,26)(22,28,25,27)(29,30)(31,37)(32,40)(33,34)(35,38)(36,41)(39,42)(43,44);(5,7,8,11)(6,9,10,12)(14,18)(16,24)(19,21)(22,25)(30,33)(31,38)(32,41)(42,43);(5,6,8,10)(7,9,11,12)(13,15)(14,19)(16,25)(17,23)(18,21)(20,28)(22,24)(26,27)(29,31)(30,35)(32,39)(33,37)(34,38)(36,42)(40,43)(41,44);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,26)(25,28)(29,32)(30,36)(31,39)(33,40)(34,41)(35,42)(37,43)(38,44);(5,7)(6,9)(8,11)(10,12)(13,16)(14,20)(15,22)(17,24)(18,26)(19,27)(21,28)(23,25)(29,33,34,30)(31,35,38,37)(32,40,41,36)(39,42,44,43)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,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, 4990200);