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 | |
1622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,8)(7,11,12,10)(13,14)(15,17)(16,21)(18,19)(20,22)(23,26)(24,28)(25,27)(29,30)(31,35)(32,34)(33,40)(36,39)(37,41)(38,43)(42,44);(2,4)(14,19)(15,22)(21,27)(23,28)(29,31,34,39)(30,35,32,36)(33,38,41,44)(37,42,40,43);(1,2,3,4)(5,6,9,8)(7,10,12,11)(13,15)(14,20)(16,23)(17,19)(18,22)(21,24)(25,28)(26,27)(29,31)(30,36)(32,35)(33,38)(34,39)(37,43)(40,42)(41,44);(5,7,9,12)(6,10,8,11)(13,16)(14,21)(15,23)(17,26)(18,25)(19,27)(20,24)(22,28)(29,32,34,30)(31,35,39,36)(33,40,41,37)(38,42,44,43);(5,8,9,6)(7,11,12,10)(13,17,18,20)(14,22,19,15)(16,24,25,26)(21,23,27,28)(29,33)(30,37)(31,38)(32,40)(34,41)(35,42)(36,43)(39,44)Polycyclic presentation:
2^9|a^2=fgi,b^2=fg,c^2=fhi,d^2=i,e^2=gi,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, 3737168);