Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1752 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)(15,18)(16,21)(17,22)(19,24)(20,26)(23,27)(25,28)(30,34)(35,37)(36,41)(43,44);(7,11)(9,12)(14,15)(19,21)(20,22)(27,28)(29,30,33,34)(31,37,39,35)(32,36,42,41)(38,44,40,43);(5,7,8,11)(6,9,10,12)(13,15,18,14)(16,21,24,19)(17,20,26,22)(23,27,25,28)(29,31,33,39)(30,35,34,37)(32,40,42,38)(36,44,41,43);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27)(29,30)(31,35)(32,41)(33,34)(36,42)(37,39)(38,44)(40,43);(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)(29,32,33,42)(30,36,34,41)(31,38,39,40)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3245372);