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 | |
1626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(6,9)(10,12)(13,14,18,19)(15,23,26,20)(16,25,27,21)(17,24,28,22)(30,34)(35,37)(36,40)(42,43);(7,11)(10,12)(14,19)(16,27)(20,23)(22,24)(29,30,33,34)(31,37,39,35)(32,36,41,40)(38,43,44,42);(5,6,8,9)(7,10,11,12)(13,15,18,26)(14,20,19,23)(16,24,27,22)(17,21,28,25)(29,30)(31,35)(32,40)(33,34)(36,41)(37,39)(38,43)(42,44);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,16,18,27)(14,21,19,25)(15,24,26,22)(17,20,28,23)(29,31)(30,35)(32,38)(33,39)(34,37)(36,42)(40,43)(41,44);(13,17,18,28)(14,22,19,24)(15,25,26,21)(16,20,27,23)(29,32,33,41)(30,36,34,40)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gh,d^2=fh,e^2=f,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, 4517404);