Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1632 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)(7,10,12,11)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,26)(25,27)(29,30,33,36)(31,37,38,34)(32,40,41,35)(39,42,44,43);(5,7,9,12)(6,10,8,11)(14,18)(19,21)(20,24)(27,28)(31,38)(32,41)(34,37)(35,40);(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,24)(17,18)(20,26)(21,23)(22,28)(25,27)(29,31,33,38)(30,34,36,37)(32,39,41,44)(35,42,40,43);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,15,17,23)(14,19,18,21)(16,25,26,22)(20,28,24,27)(29,30)(31,37)(32,35)(33,36)(34,38)(39,43)(40,41)(42,44);(5,8,9,6)(7,11,12,10)(13,16,17,26)(14,20,18,24)(15,22,23,25)(19,27,21,28)(29,32,33,41)(30,35,36,40)(31,39,38,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=fh,b^2=fgh,c^2=g,d^2=gh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4065020);