Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14,18,20)(15,21,22,16)(17,25,26,19)(23,24,28,27)(29,30,34,37)(31,38,32,35)(33,36,42,41)(39,44,40,43);(6,10)(11,12)(15,22)(16,21)(23,28)(24,27)(29,31,34,32)(30,35,37,38)(33,39,42,40)(36,43,41,44);(5,6,9,10)(7,11,8,12)(13,15)(14,16)(17,23)(18,22)(19,24)(20,21)(25,27)(26,28)(29,32,34,31)(30,35,37,38)(33,39,42,40)(36,44,41,43);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16)(14,15)(17,24)(18,21)(19,23)(20,22)(25,28)(26,27)(29,30)(31,38)(32,35)(33,41)(34,37)(36,42)(39,43)(40,44);(5,8,9,7)(6,12,10,11)(13,17,18,26)(14,19,20,25)(15,23,22,28)(16,24,21,27)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fgh,d^2=gh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3269008);