Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1708 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)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(29,30)(31,37)(32,40)(33,39)(34,35)(36,41)(38,42)(43,44);(5,6,8,9)(7,11,12,10)(15,22)(16,25)(18,21)(19,24)(30,35)(32,41)(33,37)(38,44);(6,9)(10,11)(13,15,17,22)(14,18,20,21)(16,23,25,28)(19,26,24,27)(29,31,34,39)(30,33,35,37)(32,38,41,44)(36,42,40,43);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,14)(15,21)(16,19)(17,20)(18,22)(23,27)(24,25)(26,28)(29,32)(30,36)(31,38)(33,42)(34,41)(35,40)(37,43)(39,44);(5,6)(7,10)(8,9)(11,12)(13,16,17,25)(14,19,20,24)(15,23,22,28)(18,26,21,27)(29,33)(30,31)(32,42)(34,37)(35,39)(36,38)(40,44)(41,43)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=fh,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5188306);