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 | |
1754 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)(17,24)(20,26)(21,23)(22,28)(25,27)(29,30,34,35)(31,37,39,33)(32,40,41,36)(38,42,44,43);(5,6,8,10)(7,9,11,12)(14,16)(15,21)(20,24)(22,25)(30,35)(32,41)(33,37)(38,44);(5,6)(7,9)(8,10)(11,12)(13,15,18,21)(14,19,16,23)(17,25,26,22)(20,28,24,27)(29,31,34,39)(30,33,35,37)(32,38,41,44)(36,42,40,43);(1,2,3,4)(13,16,18,14)(15,19,21,23)(17,24,26,20)(22,27,25,28)(29,32,34,41)(30,36,35,40)(31,38,39,44)(33,42,37,43);(5,7,8,11)(6,9,10,12)(13,17)(14,20)(15,22)(16,24)(18,26)(19,27)(21,25)(23,28)(29,33)(30,31)(32,42)(34,37)(35,39)(36,38)(40,44)(41,43)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=f,d^2=i,e^2=fgh,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, 4990585);