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 | |
1682 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)(5,6,9,12)(7,10,8,11)(13,14)(15,17)(16,23)(18,19)(20,25)(21,27)(22,24)(26,28)(30,31)(32,39)(36,38)(40,41);(5,7,9,8)(6,10,12,11)(14,19)(15,21)(20,23)(22,28)(29,30,34,31)(32,35,39,37)(33,36,42,38)(40,43,41,44);(5,6)(7,10)(8,11)(9,12)(13,15,18,21)(14,17,19,27)(16,22,25,28)(20,26,23,24)(29,31,34,30)(32,37,39,35)(33,38,42,36)(40,44,41,43);(1,2,3,4)(5,8,9,7)(6,11,12,10)(13,16,18,25)(14,20,19,23)(15,22,21,28)(17,26,27,24)(29,32)(30,35)(31,37)(33,41)(34,39)(36,44)(38,43)(40,42);(13,17)(14,21)(15,19)(16,24)(18,27)(20,22)(23,28)(25,26)(29,33,34,42)(30,36,31,38)(32,40,39,41)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,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, 4342354);