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 | |
1612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,8)(7,10,11,12)(13,14,18,16)(15,21,22,19)(17,20,25,24)(23,27,28,26)(29,30,34,32)(31,36,39,35)(33,40,42,37)(38,43,41,44);(7,11)(10,12)(14,16)(15,22)(17,25)(26,27)(29,31,34,39)(30,35,32,36)(33,41,42,38)(37,44,40,43);(5,7)(6,10)(8,12)(9,11)(13,15,18,22)(14,19,16,21)(17,23,25,28)(20,26,24,27)(29,31)(30,36)(32,35)(33,38)(34,39)(37,44)(40,43)(41,42);(1,2)(3,4)(5,8,9,6)(7,12,11,10)(13,16,18,14)(15,19,22,21)(17,20,25,24)(23,27,28,26)(29,32,34,30)(31,36,39,35)(33,37,42,40)(38,43,41,44);(13,17,18,25)(14,20,16,24)(15,23,22,28)(19,26,21,27)(29,33,34,42)(30,37,32,40)(31,38,39,41)(35,43,36,44)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=g,d^2=h,e^2=f,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, 2981691);