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 | |
1698 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)(5,6,9,11)(7,8,12,10)(13,14,17,18)(15,19,23,21)(16,20,25,26)(22,27,24,28)(29,30,34,37)(31,32,39,35)(33,36,41,43)(38,40,44,42);(5,7,9,12)(6,10,11,8)(14,18)(19,21)(20,26)(27,28)(33,41)(36,43)(38,44)(40,42);(7,12)(8,10)(13,15)(14,19)(16,24)(17,23)(18,21)(20,28)(22,25)(26,27)(29,31,34,39)(30,35,37,32)(33,38,41,44)(36,42,43,40);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,27)(25,26)(29,32,34,35)(30,31,37,39)(33,40,41,42)(36,38,43,44);(5,8)(6,7)(9,10)(11,12)(13,16,17,25)(14,20,18,26)(15,22,23,24)(19,27,21,28)(29,33,34,41)(30,36,37,43)(31,38,39,44)(32,40,35,42)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=g,d^2=fhi,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, 4992076);