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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
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)(31,32)(33,37)(34,35)(36,39)(38,40)(41,42)(43,44);(1,2,3,4)(14,18)(19,21)(20,24)(27,28)(30,35)(32,39)(37,44)(40,42);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,14)(15,19)(16,24)(17,18)(20,26)(21,23)(22,28)(25,27)(29,31,34,36)(30,32,35,39)(33,41,43,38)(37,42,44,40);(5,7,9,11)(6,10,8,12)(13,15)(14,19)(16,25)(17,23)(18,21)(20,28)(22,26)(24,27)(29,32)(30,36)(31,35)(33,42)(34,39)(37,38)(40,43)(41,44);(5,8,9,6)(7,12,11,10)(13,16,17,26)(14,20,18,24)(15,22,23,25)(19,27,21,28)(29,33,34,43)(30,37,35,44)(31,38,36,41)(32,40,39,42)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gi,d^2=fi,e^2=i,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, 3701969);