Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1674 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)(5,6)(7,11)(8,9)(10,12)(13,14,18,16)(15,21,23,19)(17,20,25,24)(22,27,28,26)(29,30)(31,38)(32,33)(34,35)(36,41)(37,42)(39,44)(40,43);(5,7,9,12)(6,10,8,11)(17,25)(20,24)(22,28)(26,27)(30,35)(33,42)(36,38)(40,44);(5,6,9,8)(7,11,12,10)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27)(29,31,34,41)(30,36,35,38)(32,39,37,43)(33,40,42,44);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(13,16,18,14)(15,19,23,21)(17,24,25,20)(22,26,28,27)(29,32)(30,33)(31,39)(34,37)(35,42)(36,40)(38,44)(41,43);(5,7)(6,11)(8,10)(9,12)(13,17,18,25)(14,20,16,24)(15,22,23,28)(19,26,21,27)(29,33)(30,37)(31,40)(32,35)(34,42)(36,43)(38,39)(41,44)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=g,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5524124);