Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| 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 | |
1758 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)(5,6,8,10)(7,9,11,12)(13,14,18,15)(16,21,23,19)(17,20,26,22)(24,28,25,27)(30,35)(31,36)(33,41)(39,42);(7,11)(9,12)(14,15)(16,23)(20,22)(24,25)(29,30,34,35)(31,32,36,40)(33,37,41,43)(38,39,44,42);(5,7,8,11)(6,9,10,12)(13,15,18,14)(16,19,23,21)(17,22,26,20)(24,27,25,28)(29,31)(30,32)(33,38)(34,36)(35,40)(37,39)(41,44)(42,43);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,23)(14,19,15,21)(17,25,26,24)(20,28,22,27)(29,32)(30,36)(31,35)(33,42)(34,40)(37,38)(39,41)(43,44);(13,17,18,26)(14,20,15,22)(16,24,23,25)(19,27,21,28)(29,33,34,41)(30,37,35,43)(31,38,36,44)(32,39,40,42)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2643947);