Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| 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 | |
1685 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)(7,11)(10,12)(13,14)(15,18)(16,21)(17,22)(19,24)(20,25)(23,26)(27,28)(29,30,34,38)(31,35,36,32)(33,41,42,37)(39,40,44,43);(6,9)(10,12)(14,15)(19,21)(20,22)(26,27)(29,31,34,36)(30,32,38,35)(33,39,42,44)(37,40,41,43);(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27)(29,31)(30,35)(32,38)(33,39)(34,36)(37,43)(40,41)(42,44);(1,2,3,4)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,32,34,35)(30,36,38,31)(33,40,42,43)(37,44,41,39);(5,7)(6,10)(8,11)(9,12)(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)(29,33,34,42)(30,37,38,41)(31,39,36,44)(32,40,35,43)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=g,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4371267);