Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
1738 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)(7,10)(8,12)(9,11)(14,18)(16,24)(19,21)(22,28)(29,30,33,34)(31,37,39,35)(32,36,40,41)(38,43,44,42);(5,7,9,12)(6,10,11,8)(14,18)(19,21)(20,25)(26,27)(30,34)(32,40)(35,37)(38,44);(5,8)(6,7)(9,10)(11,12)(13,14,17,18)(15,21,23,19)(16,20,24,25)(22,27,28,26)(29,30)(31,35)(32,36)(33,34)(37,39)(38,42)(40,41)(43,44);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,31)(30,35)(32,38)(33,39)(34,37)(36,42)(40,44)(41,43);(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)(29,32,33,40)(30,36,34,41)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=fh,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4851434);