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 | |
1782 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)(5,6,10,8)(7,11,9,12)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(29,30,33,34)(31,37,38,35)(32,40,41,36)(39,43,42,44);(5,7,10,9)(6,11,8,12)(15,22)(18,21)(23,28)(26,27)(30,34)(31,38)(32,41)(43,44);(5,8,10,6)(7,11,9,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28)(29,31,33,38)(30,35,34,37)(32,42,41,39)(36,44,40,43);(1,2,3,4)(5,9,10,7)(6,11,8,12)(13,15)(14,18)(16,23)(17,22)(19,26)(20,21)(24,27)(25,28)(29,32,33,41)(30,36,34,40)(31,39,38,42)(35,43,37,44);(13,16,17,25)(14,19,20,24)(15,23,22,28)(18,26,21,27)(29,30)(31,35)(32,36)(33,34)(37,38)(39,43)(40,41)(42,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=h,d^2=fgi,e^2=fg,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, 1652914);