Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1911 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)(5,6)(7,11)(8,10)(9,12)(13,14,17,18)(15,21,23,19)(16,20,24,26)(22,28,25,27)(31,37)(32,39)(34,40)(35,42);(5,6)(7,11)(8,10)(9,12)(14,18)(16,24)(19,21)(22,25)(29,30,33,36)(31,34,37,40)(32,35,39,42)(38,41,43,44);(5,6,8,10)(7,9,12,11)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,26)(25,28)(29,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,23)(14,19,18,21)(16,25,24,22)(20,28,26,27)(29,31,33,37)(30,34,36,40)(32,38,39,43)(35,41,42,44);(5,7,8,12)(6,9,10,11)(13,16,17,24)(14,20,18,26)(15,22,23,25)(19,27,21,28)(29,32)(30,35)(31,38)(33,39)(34,41)(36,42)(37,43)(40,44)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=fg,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5524144);