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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(10,12)(13,14,17,18)(15,21,22,19)(16,20,24,26)(23,28,25,27)(29,30,34,37)(31,38,32,35)(33,41,42,36)(39,43,40,44);(2,4)(14,18)(15,22)(16,24)(27,28)(29,31,34,32)(30,35,37,38)(33,40,42,39)(36,44,41,43);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,14)(15,21)(16,20)(17,18)(19,22)(23,28)(24,26)(25,27)(29,32,34,31)(30,35,37,38)(33,40,42,39)(36,43,41,44);(5,7)(6,10)(8,12)(9,11)(13,15,17,22)(14,19,18,21)(16,25,24,23)(20,28,26,27)(29,30)(31,35)(32,38)(33,41)(34,37)(36,42)(39,44)(40,43);(5,8,9,6)(7,12,11,10)(13,16,17,24)(14,20,18,26)(15,23,22,25)(19,27,21,28)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=gi,d^2=g,e^2=fgi,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, 3142588);