Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
1702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,7)(8,11,12,10)(13,14,18,15)(16,19,24,21)(17,20,26,22)(23,27,25,28)(29,30,34,36)(31,37,38,32)(33,41,42,35)(39,40,44,43);(2,4)(5,6,9,7)(8,10,12,11)(14,15)(19,21)(20,22)(27,28)(31,38)(32,37)(39,44)(40,43);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,15,18,14)(16,21,24,19)(17,22,26,20)(23,28,25,27)(29,31)(30,32)(33,39)(34,38)(35,40)(36,37)(41,43)(42,44);(5,8)(6,10)(7,11)(9,12)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27)(29,32)(30,31)(33,40)(34,37)(35,39)(36,38)(41,44)(42,43);(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)(29,33,34,42)(30,35,36,41)(31,39,38,44)(32,40,37,43)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=ghi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2554022);