Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1786 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)(10,12)(13,14,17,18)(15,19,21,26)(16,23,25,20)(22,28,24,27)(30,35)(32,39)(37,43)(40,41);(6,9)(10,12)(14,18)(15,21)(20,23)(22,24)(29,30,34,35)(31,32,36,39)(33,37,42,43)(38,40,44,41);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18)(21,26)(22,27)(23,25)(24,28)(29,31,34,36)(30,32,35,39)(33,38,42,44)(37,40,43,41);(1,2)(3,4)(13,15,17,21)(14,19,18,26)(16,24,25,22)(20,28,23,27)(29,32)(30,36)(31,35)(33,41)(34,39)(37,38)(40,42)(43,44);(5,7)(6,10)(8,11)(9,12)(13,16,17,25)(14,20,18,23)(15,22,21,24)(19,27,26,28)(29,33,34,42)(30,37,35,43)(31,38,36,44)(32,40,39,41)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=fg,d^2=g,e^2=f,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, 4774047);