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 | |
1730 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)(7,11)(9,12)(13,14,17,18)(15,21,23,19)(16,24,26,20)(22,27,25,28)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(5,6,8,10)(7,9,11,12)(14,18)(19,21)(20,24)(27,28)(32,41)(35,40)(38,44)(42,43);(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,24)(17,18)(20,26)(21,23)(22,28)(25,27)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,19)(16,25)(17,23)(18,21)(20,28)(22,26)(24,27)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43);(13,16,17,26)(14,20,18,24)(15,22,23,25)(19,27,21,28)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,41)(43,44)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fg,d^2=hi,e^2=f,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, 3376415);