Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1938 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)(14,18)(15,21)(16,24)(26,28)(29,30,34,35)(31,32,37,40)(33,36,43,44)(38,39,41,42);(5,6,8,10)(7,9,11,12)(16,24)(20,25)(23,27)(26,28)(30,35)(31,37)(36,44)(38,41);(5,6)(7,9)(8,10)(11,12)(13,14,17,18)(15,22,21,19)(16,20,24,25)(23,28,27,26)(29,31)(30,32)(33,41)(34,37)(35,40)(36,42)(38,43)(39,44);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15,17,21)(14,19,18,22)(16,23,24,27)(20,26,25,28)(29,32)(30,31)(33,42)(34,40)(35,37)(36,41)(38,44)(39,43);(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)(29,33,34,43)(30,36,35,44)(31,38,37,41)(32,39,40,42)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=g,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4048084);