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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,8)(9,11)(10,12)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26)(29,30,34,35)(31,32,39,36)(33,41,42,37)(38,44,43,40);(5,7,9,12)(6,10,11,8)(15,21)(18,25)(22,27)(26,28)(30,35)(32,36)(33,42)(38,43);(7,12)(8,10)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27)(29,31)(30,36)(32,35)(33,43)(34,39)(37,40)(38,42)(41,44);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28)(29,32,34,36)(30,31,35,39)(33,40,42,44)(37,38,41,43);(5,8)(6,7)(9,10)(11,12)(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)(29,33,34,42)(30,37,35,41)(31,38,39,43)(32,40,36,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,d^2=fghi,e^2=fh,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, 5191862);