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 | |
1822 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)(5,6,8,10)(7,11,12,9)(15,21)(16,23)(19,24)(20,25)(29,30)(31,36)(32,40)(33,38)(34,35)(37,43)(39,42)(41,44);(5,7,8,12)(6,9,10,11)(14,18)(19,24)(20,25)(26,28)(30,35)(33,36)(37,40)(42,44);(6,10)(9,11)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,34,38)(30,36,35,33)(32,41,43,39)(37,42,40,44);(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,43)(30,37,35,40)(31,39,38,41)(33,42,36,44);(5,7)(6,11)(8,12)(9,10)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,36)(30,38,35,31)(32,42,43,44)(37,41,40,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fg,d^2=ghi,e^2=fgh,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, 4410256);