Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1802 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(9,12)(13,14)(15,19)(16,18)(17,20)(21,27)(22,23)(24,26)(25,28)(29,30,33,34)(31,37,38,35)(32,40,41,36)(39,42,44,43);(5,6,8,10)(7,9,11,12)(14,16)(19,23)(20,24)(27,28)(30,34)(31,38)(32,41)(42,43);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,23)(17,25)(18,22)(20,28)(21,26)(24,27)(29,31,33,38)(30,35,34,37)(32,39,41,44)(36,42,40,43);(1,2,3,4)(13,16,18,14)(15,19,22,23)(17,24,26,20)(21,27,25,28)(29,32,33,41)(30,36,34,40)(31,39,38,44)(35,42,37,43);(5,7,8,11)(6,9,10,12)(13,17,18,26)(14,20,16,24)(15,21,22,25)(19,27,23,28)(29,30)(31,37)(32,36)(33,34)(35,38)(39,43)(40,41)(42,44)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 4990499);