Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14,18,19)(15,20,23,27)(16,24,26,17)(21,22,25,28)(29,30)(31,36)(32,39)(33,35)(34,37)(38,43)(40,44)(41,42);(6,9)(10,12)(14,19)(17,24)(20,27)(22,28)(29,31,35,32)(30,36,33,39)(34,38,42,40)(37,43,41,44);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,22)(18,23)(19,27)(21,26)(24,28)(29,32,35,31)(30,36,33,39)(34,40,42,38)(37,43,41,44);(1,2,3,4)(13,16)(14,17)(15,21)(18,26)(19,24)(20,28)(22,27)(23,25)(29,33,35,30)(31,36,32,39)(34,41,42,37)(38,43,40,44);(5,7,8,11)(6,10,9,12)(13,17)(14,16)(15,22)(18,24)(19,26)(20,25)(21,27)(23,28)(29,34,35,42)(30,37,33,41)(31,38,32,40)(36,43,39,44)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=fi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4408565);