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 | |
1598 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)(5,6,9,7)(8,11,12,10)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25)(29,30)(31,32)(33,41)(34,35)(36,39)(37,43)(38,44)(40,42);(5,6,9,7)(8,11,12,10)(16,24)(19,26)(22,28)(25,27)(30,35)(32,39)(37,41)(40,44);(6,7)(10,11)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27)(29,31)(30,32)(33,42)(34,36)(35,39)(37,44)(38,43)(40,41);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28)(29,32)(30,36)(31,35)(33,40)(34,39)(37,42)(38,41)(43,44);(5,8,9,12)(6,10,7,11)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)(29,33,34,43)(30,37,35,41)(31,38,36,42)(32,40,39,44)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=fgi,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, 5188371);