Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1666 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)(10,12)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(29,30)(31,33)(32,39)(34,35)(36,41)(37,43)(38,40)(42,44);(5,6,9,8)(7,10,11,12)(15,22)(18,21)(23,28)(26,27)(30,35)(31,37)(36,39)(38,44);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,23)(17,22)(19,26)(20,21)(24,27)(25,28)(29,31,34,37)(30,33,35,43)(32,38,41,44)(36,42,39,40);(1,2,3,4)(5,8,9,6)(7,12,11,10)(13,14)(15,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28)(29,32,34,41)(30,36,35,39)(31,38,37,44)(33,42,43,40);(13,16,17,25)(14,19,20,24)(15,23,22,28)(18,26,21,27)(29,33)(30,37)(31,35)(32,40)(34,43)(36,38)(39,44)(41,42)Polycyclic presentation:
2^9|a^2=fh,b^2=fgh,c^2=f,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4342515);