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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1799 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)(7,12)(8,9)(10,11)(13,14,17,18)(15,21,23,19)(16,20,24,25)(22,27,28,26)(29,30,34,37)(31,35,32,38)(33,36,41,43)(39,42,40,44);(6,11)(7,9)(14,18)(16,24)(19,21)(22,28)(29,31,34,32)(30,35,37,38)(33,39,41,40)(36,42,43,44);(5,7,10,9)(6,12,11,8)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,32,34,31)(30,35,37,38)(33,40,41,39)(36,42,43,44);(1,2)(3,4)(5,8,10,12)(6,7,11,9)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,30)(31,38)(32,35)(33,36)(34,37)(39,44)(40,42)(41,43);(5,9,10,7)(6,12,11,8)(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)(29,33,34,41)(30,36,37,43)(31,39,32,40)(35,42,38,44)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=gh,d^2=h,e^2=fg,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, 4377278);