Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,8)(9,10)(11,12)(14,18)(16,23)(19,24)(22,27)(30,34)(31,37)(36,42)(39,40);(6,10)(7,12)(14,18)(15,21)(20,25)(22,27)(29,30,33,34)(31,38,37,35)(32,36,41,42)(39,44,40,43);(6,10)(8,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,33,37)(30,35,34,38)(32,40,41,39)(36,44,42,43);(1,2)(3,4)(5,7,9,12)(6,11,10,8)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,30)(31,35)(32,36)(33,34)(37,38)(39,43)(40,44)(41,42);(5,8,9,11)(6,12,10,7)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,32,33,41)(30,36,34,42)(31,39,37,40)(35,43,38,44)Polycyclic presentation:
2^9|a^2=fi,b^2=fgh,c^2=h,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4517494);