Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1876 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)(5,6,8,10)(7,9,11,12)(13,14)(15,16)(17,22)(18,19)(20,21)(23,24)(25,28)(26,27)(29,30)(31,35)(32,38)(33,41)(34,37)(36,42)(39,44)(40,43);(7,11)(9,12)(14,19)(15,21)(22,27)(23,28)(29,31,34,32)(30,35,37,38)(33,39,42,40)(36,43,41,44);(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,19)(17,23)(18,21)(22,25)(24,27)(26,28)(29,32,34,31)(30,35,37,38)(33,40,42,39)(36,43,41,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,20)(14,21,19,15)(17,25,26,24)(22,23,27,28)(29,30)(31,38)(32,35)(33,36)(34,37)(39,44)(40,43)(41,42);(13,17,18,26)(14,22,19,27)(15,23,21,28)(16,24,20,25)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,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, 2668040);