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 | |
1773 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)(6,7)(8,12)(13,14,17,20)(15,21,22,18)(16,19,24,26)(23,27,28,25)(30,35)(32,37)(33,41)(38,44);(6,7)(10,11)(15,22)(18,21)(23,28)(25,27)(29,30,34,35)(31,37,39,32)(33,42,41,36)(38,40,44,43);(5,6,9,7)(8,10,12,11)(13,14)(15,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28)(29,31,34,39)(30,32,35,37)(33,38,41,44)(36,40,42,43);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,22)(14,18,20,21)(16,23,24,28)(19,25,26,27)(29,32)(30,31)(33,40)(34,37)(35,39)(36,38)(41,43)(42,44);(5,8)(6,10)(7,11)(9,12)(13,16,17,24)(14,19,20,26)(15,23,22,28)(18,25,21,27)(29,33,34,41)(30,36,35,42)(31,38,39,44)(32,40,37,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=h,d^2=g,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, 4381167);