Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,8)(7,11,12,10)(13,14,18,19)(15,22,20,16)(17,25,28,21)(23,24,26,27)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(2,4)(14,19)(15,20)(21,25)(23,26)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,15,18,20)(14,16,19,22)(17,26,28,23)(21,27,25,24)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,41)(43,44);(5,7,9,12)(6,10,8,11)(13,16)(14,20)(15,19)(17,27)(18,22)(21,23)(24,28)(25,26)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(5,8,9,6)(7,11,12,10)(13,17)(14,21)(15,23)(16,24)(18,28)(19,25)(20,26)(22,27)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=fgi,d^2=fi,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3732679);