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 | |
1769 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)(13,14,18,15)(16,21,24,19)(17,22,25,20)(23,26,28,27)(29,30,33,36)(31,37,39,34)(32,40,42,35)(38,43,41,44);(6,9)(10,12)(14,15)(17,25)(19,21)(23,28)(29,31,33,39)(30,34,36,37)(32,41,42,38)(35,44,40,43);(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27)(29,31)(30,34)(32,38)(33,39)(35,43)(36,37)(40,44)(41,42);(1,2,3,4)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,44)(40,42)(41,43);(5,7,8,11)(6,10,9,12)(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)(29,32)(30,35)(31,38)(33,42)(34,43)(36,40)(37,44)(39,41)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fg,d^2=gi,e^2=gh,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, 4408689);