Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1604 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)(5,6,9,12)(7,10,11,8)(14,18)(15,21)(20,25)(22,27)(29,30,34,36)(31,37,40,35)(32,41,43,33)(38,44,42,39);(5,7)(6,8)(9,11)(10,12)(15,21)(19,24)(22,27)(26,28)(29,31,34,40)(30,35,36,37)(32,42,43,38)(33,39,41,44);(5,7)(6,10)(8,12)(9,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31)(30,35)(32,38)(33,44)(34,40)(36,37)(39,41)(42,43);(1,2,3,4)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32)(30,33)(31,38)(34,43)(35,44)(36,41)(37,39)(40,42);(5,8,9,10)(6,11,12,7)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33)(30,32)(31,39)(34,41)(35,42)(36,43)(37,38)(40,44)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=gh,d^2=hi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4575661);