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 | |
1706 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)(13,14,18,19)(15,16,20,24)(17,25,26,21)(22,27,28,23)(29,30,34,38)(31,36,32,35)(33,37,42,41)(39,44,40,43);(6,10)(11,12)(14,19)(15,20)(17,26)(23,27)(29,31,34,32)(30,35,38,36)(33,40,42,39)(37,44,41,43);(5,6,9,10)(7,11,8,12)(13,15)(14,16)(17,22)(18,20)(19,24)(21,23)(25,27)(26,28)(29,32,34,31)(30,36,38,35)(33,40,42,39)(37,44,41,43);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,19)(17,27)(18,24)(21,22)(23,26)(25,28)(29,30)(31,36)(32,35)(33,41)(34,38)(37,42)(39,43)(40,44);(5,8,9,7)(6,12,10,11)(13,17,18,26)(14,21,19,25)(15,22,20,28)(16,23,24,27)(29,33,34,42)(30,37,38,41)(31,39,32,40)(35,43,36,44)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=fgh,e^2=gh,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, 2991670);