Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(10,12)(13,14,17,18)(15,21,23,19)(16,24,26,20)(22,27,25,28)(29,30)(31,35)(32,36)(33,34)(37,42)(38,43)(39,44)(40,41);(1,2,3,4)(6,9)(10,12)(14,18)(19,21)(20,24)(27,28)(30,34)(31,37)(36,40)(38,39);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,24)(17,18)(20,26)(21,23)(22,28)(25,27)(29,31,33,37)(30,35,34,42)(32,39,41,38)(36,44,40,43);(5,7)(6,10)(8,11)(9,12)(13,15,17,23)(14,19,18,21)(16,25,26,22)(20,28,24,27)(29,30)(31,35)(32,40)(33,34)(36,41)(37,42)(38,44)(39,43);(13,16,17,26)(14,20,18,24)(15,22,23,25)(19,27,21,28)(29,32)(30,36)(31,38)(33,41)(34,40)(35,43)(37,39)(42,44)Polycyclic presentation:
2^9|a^2=g,b^2=fghi,c^2=fi,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3341135);