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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1626 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)(5,6,8,10)(7,11,12,9)(13,14)(15,21)(16,18)(17,20)(19,23)(22,27)(24,25)(26,28)(29,30,34,33)(31,37,39,35)(32,40,41,36)(38,42,44,43);(5,6,8,10)(7,9,12,11)(14,16)(17,25)(19,21)(22,28)(32,41)(36,40)(38,44)(42,43);(5,6)(7,9)(8,10)(11,12)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27)(29,31,34,39)(30,35,33,37)(32,38,41,44)(36,42,40,43);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,14)(15,19,23,21)(17,24,25,20)(22,26,28,27)(29,32)(30,36)(31,38)(33,40)(34,41)(35,42)(37,43)(39,44);(5,7,8,12)(6,9,10,11)(13,17,18,25)(14,20,16,24)(15,22,23,28)(19,26,21,27)(29,33,34,30)(31,37,39,35)(32,40,41,36)(38,43,44,42)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=gh,d^2=f,e^2=g,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, 4375029);