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 | |
1810 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)(7,12)(8,11)(9,10)(13,14,18,21)(15,16,23,19)(17,20,25,27)(22,24,28,26)(29,30,34,38)(31,35,36,32)(33,41,43,37)(39,40,42,44);(6,10)(7,11)(17,25)(20,27)(22,28)(24,26)(29,31,34,36)(30,32,38,35)(33,39,43,42)(37,40,41,44);(7,11)(8,12)(13,15,18,23)(14,19,21,16)(17,22,25,28)(20,26,27,24)(29,31)(30,35)(32,38)(33,42)(34,36)(37,40)(39,43)(41,44);(1,2,3,4)(5,7,9,11)(6,8,10,12)(13,16,18,19)(14,15,21,23)(17,24,25,26)(20,22,27,28)(29,32,34,35)(30,36,38,31)(33,40,43,44)(37,42,41,39);(5,8)(6,11)(7,10)(9,12)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)(29,33,34,43)(30,37,38,41)(31,39,36,42)(32,40,35,44)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=i,e^2=h,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, 4990947);