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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,26)(27,28)(29,30,34,32)(31,37,38,35)(33,40,41,36)(39,42,44,43);(6,9)(10,12)(14,18)(16,25)(19,21)(22,28)(30,32)(31,38)(33,41)(42,43);(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,31,34,38)(30,35,32,37)(33,39,41,44)(36,42,40,43);(1,2)(3,4)(13,15,17,23)(14,19,18,21)(16,22,25,28)(20,26,24,27)(29,32,34,30)(31,35,38,37)(33,40,41,36)(39,42,44,43);(5,7)(6,10)(8,11)(9,12)(13,16,17,25)(14,20,18,24)(15,22,23,28)(19,26,21,27)(29,33)(30,36)(31,39)(32,40)(34,41)(35,42)(37,43)(38,44)Polycyclic presentation:
2^9|a^2=g,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^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4373220);