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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1834 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)(5,6,9,10)(7,11,12,8)(13,14)(15,23)(16,26)(17,25)(18,19)(20,24)(21,27)(22,28)(31,37)(32,39)(34,40)(35,42);(6,10)(7,12)(14,19)(15,24)(16,27)(17,28)(29,30,33,36)(31,34,37,40)(32,35,39,42)(38,41,43,44);(7,12)(8,11)(13,15)(14,20)(16,25)(17,21)(18,24)(19,23)(22,27)(26,28)(29,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,44);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,16)(14,21)(15,25)(17,20)(18,27)(19,26)(22,24)(23,28)(29,31)(30,34)(32,38)(33,37)(35,41)(36,40)(39,43)(42,44);(5,8)(6,7)(9,11)(10,12)(13,17)(14,22)(15,26)(16,20)(18,28)(19,25)(21,24)(23,27)(29,32,33,39)(30,35,36,42)(31,38,37,43)(34,41,40,44)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,d^2=fghi,e^2=gh,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, 4992144);