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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1954 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)(15,19)(16,18)(17,20)(21,27)(22,23)(24,26)(25,28)(29,30,35,38)(31,34,32,36)(33,41,43,37)(39,44,40,42);(6,9)(10,11)(14,16)(19,23)(20,24)(27,28)(29,31,35,32)(30,36,38,34)(33,40,43,39)(37,42,41,44);(5,6,8,9)(7,11,12,10)(13,15,18,22)(14,19,16,23)(17,25,26,21)(20,28,24,27)(29,32,35,31)(30,34,38,36)(33,40,43,39)(37,42,41,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,19,22,23)(17,24,26,20)(21,27,25,28)(29,33,35,43)(30,37,38,41)(31,39,32,40)(34,42,36,44);(5,7)(6,10)(8,12)(9,11)(13,17,18,26)(14,20,16,24)(15,21,22,25)(19,27,23,28)(29,34)(30,32)(31,38)(33,42)(35,36)(37,40)(39,41)(43,44)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=f,d^2=hi,e^2=fgh,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, 4991670);