Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(7,11)(9,12)(14,18)(15,21)(20,25)(22,27)(29,30,34,35)(31,37,40,36)(32,41,42,33)(38,39,44,43);(5,6,8,10)(7,9,11,12)(15,21)(19,24)(22,27)(26,28)(30,35)(32,42)(36,37)(38,44);(5,6)(7,9)(8,10)(11,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,34,40)(30,36,35,37)(32,38,42,44)(33,39,41,43);(1,2)(3,4)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,42)(30,33,35,41)(31,38,40,44)(36,39,37,43);(5,7)(6,9)(8,11)(10,12)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33)(30,32)(31,39)(34,41)(35,42)(36,38)(37,44)(40,43)Polycyclic presentation:
2^9|a^2=fgi,b^2=h,c^2=g,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4774754);