Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1681 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)(5,6,9,8)(7,10,11,12)(13,14,17,18)(15,21,22,19)(16,24,25,20)(23,26,28,27)(30,32)(35,39)(36,40)(42,43);(7,11)(10,12)(14,18)(15,22)(16,25)(26,27)(29,30,34,32)(31,35,38,39)(33,40,41,36)(37,43,44,42);(5,7,9,11)(6,10,8,12)(13,15,17,22)(14,19,18,21)(16,23,25,28)(20,26,24,27)(29,31)(30,35)(32,39)(33,37)(34,38)(36,42)(40,43)(41,44);(1,2,3,4)(13,14)(15,21)(16,20)(17,18)(19,22)(23,27)(24,25)(26,28)(29,32,34,30)(31,35,38,39)(33,40,41,36)(37,42,44,43);(5,8,9,6)(7,12,11,10)(13,16,17,25)(14,20,18,24)(15,23,22,28)(19,26,21,27)(29,33,34,41)(30,36,32,40)(31,37,38,44)(35,42,39,43)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=h,d^2=gi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1555506);