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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1893 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)(5,6,8,9)(7,11,12,10)(13,14)(15,21)(16,18)(17,20)(19,22)(23,28)(24,25)(26,27)(29,30,34,37)(31,35,32,39)(33,36,41,44)(38,42,40,43);(6,9)(7,12)(14,16)(15,22)(17,25)(27,28)(29,31,34,32)(30,35,37,39)(33,40,41,38)(36,43,44,42);(6,9)(10,11)(13,15)(14,19)(16,21)(17,26)(18,22)(20,28)(23,25)(24,27)(29,32,34,31)(30,35,37,39)(33,40,41,38)(36,42,44,43);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,16,18,14)(15,21,22,19)(17,24,25,20)(23,28,26,27)(29,30)(31,35)(32,39)(33,36)(34,37)(38,42)(40,43)(41,44);(5,6)(7,10)(8,9)(11,12)(13,17,18,25)(14,20,16,24)(15,23,22,26)(19,27,21,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,44)(39,43)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fg,d^2=ghi,e^2=h,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, 5192001);