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 | |
1695 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)(13,14)(15,21)(16,18)(17,20)(19,22)(23,27)(24,25)(26,28)(29,30,33,37)(31,35,39,34)(32,40,42,36)(38,43,41,44);(6,10)(11,12)(14,16)(15,22)(17,25)(26,27)(29,31,33,39)(30,34,37,35)(32,41,42,38)(36,44,40,43);(5,6,9,10)(7,11,8,12)(13,15)(14,19)(16,21)(17,23)(18,22)(20,26)(24,27)(25,28)(29,31)(30,35)(32,38)(33,39)(34,37)(36,44)(40,43)(41,42);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16,18,14)(15,19,22,21)(17,20,25,24)(23,27,28,26)(29,30)(31,34)(32,40)(33,37)(35,39)(36,42)(38,44)(41,43);(5,8,9,7)(6,12,10,11)(13,17,18,25)(14,20,16,24)(15,23,22,28)(19,26,21,27)(29,32)(30,36)(31,38)(33,42)(34,43)(35,44)(37,40)(39,41)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fh,d^2=fgh,e^2=fgh,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, 2991720);