Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(6,9)(10,12)(13,14,17,20)(15,21,22,18)(16,19,24,26)(23,27,28,25)(29,30,34,37)(31,38,32,35)(33,41,42,36)(39,43,40,44);(2,4)(15,22)(16,24)(18,21)(19,26)(29,31,34,32)(30,35,37,38)(33,40,42,39)(36,44,41,43);(1,2,3,4)(13,15)(14,18)(16,23)(17,22)(19,25)(20,21)(24,28)(26,27)(29,32,34,31)(30,35,37,38)(33,40,42,39)(36,43,41,44);(5,6,8,9)(7,10,11,12)(13,16,17,24)(14,19,20,26)(15,23,22,28)(18,25,21,27)(29,30)(31,35)(32,38)(33,41)(34,37)(36,42)(39,44)(40,43);(5,7,8,11)(6,10,9,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fgh,d^2=fgi,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3178920);