Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,10)(7,9,11,12)(13,14,18,15)(16,21,23,19)(17,20,26,22)(24,28,25,27)(30,31)(33,41)(35,37)(39,42);(2,4)(14,15)(16,23)(20,22)(24,25)(29,30,34,31)(32,35,40,37)(33,36,41,38)(39,43,42,44);(1,2)(3,4)(13,15,18,14)(16,21,23,19)(17,20,26,22)(24,27,25,28)(29,31,34,30)(32,37,40,35)(33,36,41,38)(39,43,42,44);(5,6)(7,9)(8,10)(11,12)(13,16)(14,19)(15,21)(17,25)(18,23)(20,28)(22,27)(24,26)(29,32,34,40)(30,35,31,37)(33,42,41,39)(36,44,38,43);(5,7,8,11)(6,9,10,12)(13,17)(14,20)(15,22)(16,24)(18,26)(19,27)(21,28)(23,25)(29,33)(30,36)(31,38)(32,39)(34,41)(35,43)(37,44)(40,42)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,d^2=g,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3536701);