Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| 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 | |
1738 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)(6,9)(7,11)(16,23)(20,25)(21,27)(26,28)(29,30,34,33)(31,37,39,35)(32,36,40,41)(38,43,44,42);(2,4)(5,6,8,9)(7,10,11,12)(14,18)(19,24)(20,25)(26,28)(32,40)(36,41)(38,44)(42,43);(2,4)(5,7)(6,10)(8,11)(9,12)(13,14,17,18)(15,19,22,24)(16,20,23,25)(21,26,27,28)(29,31,34,39)(30,35,33,37)(32,38,40,44)(36,42,41,43);(1,2)(3,4)(5,6)(7,12)(8,9)(10,11)(13,15,17,22)(14,19,18,24)(16,21,23,27)(20,26,25,28)(29,32)(30,36)(31,38)(33,41)(34,40)(35,42)(37,43)(39,44);(13,16)(14,20)(15,21)(17,23)(18,25)(19,26)(22,27)(24,28)(29,33,34,30)(31,37,39,35)(32,41,40,36)(38,43,44,42)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=fgh,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4362253);