Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1590 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)(6,9)(7,11)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26)(29,30,35,40)(31,37,32,36)(33,42,43,38)(34,41,44,39);(5,6)(7,10)(8,9)(11,12)(15,21)(18,25)(22,27)(26,28)(29,31,35,32)(30,36,40,37)(33,39,43,41)(34,42,44,38);(7,11)(10,12)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27)(29,32,35,31)(30,37,40,36)(33,39,43,41)(34,42,44,38);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28)(29,33)(30,38)(31,41)(32,39)(34,37)(35,43)(36,44)(40,42);(5,7)(6,10)(8,11)(9,12)(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)(29,34)(30,39)(31,42)(32,38)(33,37)(35,44)(36,43)(40,41)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=fhi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4579076);