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 | |
1712 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)(5,6,8,11)(7,10,12,9)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26)(29,30,34,33)(31,37,38,35)(32,40,41,36)(39,42,44,43);(5,7,8,12)(6,9,11,10)(15,21)(18,25)(22,27)(26,28)(30,33)(31,38)(32,41)(42,43);(5,7,8,12)(6,10,11,9)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27)(29,31,34,38)(30,35,33,37)(32,39,41,44)(36,42,40,43);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28)(29,32,34,41)(30,36,33,40)(31,39,38,44)(35,42,37,43);(5,6)(7,9)(8,11)(10,12)(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)(29,33,34,30)(31,35,38,37)(32,40,41,36)(39,42,44,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=f,d^2=fhi,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, 5191626);