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 | |
1884 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)(9,12)(13,14)(15,21)(16,18)(17,20)(19,23)(22,28)(24,25)(26,27)(29,30,34,37)(31,33,38,35)(32,40,43,36)(39,44,41,42);(5,6,8,10)(7,9,11,12)(14,16)(17,25)(19,21)(22,26)(31,38)(33,35)(39,41)(42,44);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,21)(17,26)(18,23)(20,28)(22,25)(24,27)(29,31)(30,35)(32,41)(33,37)(34,38)(36,42)(39,43)(40,44);(1,2,3,4)(13,16,18,14)(15,21,23,19)(17,24,25,20)(22,28,26,27)(29,32,34,43)(30,36,37,40)(31,39,38,41)(33,42,35,44);(5,7,8,11)(6,9,10,12)(13,17,18,25)(14,20,16,24)(15,22,23,26)(19,27,21,28)(29,33,34,35)(30,31,37,38)(32,42,43,44)(36,39,40,41)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,d^2=i,e^2=fh,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, 4990374);