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 | |
1730 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)(7,11)(10,12)(13,14,18,21)(15,22,19,16)(17,25,26,20)(23,24,27,28)(30,34)(31,37)(32,40)(42,44);(6,9)(10,12)(15,19)(16,22)(23,27)(24,28)(29,30,33,34)(31,38,37,35)(32,41,40,36)(39,42,43,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,19)(14,16,21,22)(17,23,26,27)(20,24,25,28)(29,30)(31,38)(32,36)(33,34)(35,37)(39,44)(40,41)(42,43);(1,2)(3,4)(13,16)(14,19)(15,21)(17,24)(18,22)(20,27)(23,25)(26,28)(29,31,33,37)(30,35,34,38)(32,39,40,43)(36,42,41,44);(5,7)(6,10)(8,11)(9,12)(13,17,18,26)(14,20,21,25)(15,23,19,27)(16,24,22,28)(29,32,33,40)(30,36,34,41)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fg,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, 4373211);