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 | |
1606 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)(7,11)(10,12)(14,18)(15,21)(20,25)(22,27)(29,30,35,40)(31,37,32,36)(33,42,43,38)(34,41,44,39);(6,9)(10,12)(15,21)(19,24)(22,27)(26,28)(29,31,35,32)(30,36,40,37)(33,39,43,41)(34,42,44,38);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,32,35,31)(30,37,40,36)(33,39,43,41)(34,42,44,38);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,33)(30,38)(31,41)(32,39)(34,37)(35,43)(36,44)(40,42);(5,7)(6,10)(8,11)(9,12)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,34)(30,39)(31,42)(32,38)(33,37)(35,44)(36,43)(40,41)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=f,d^2=ghi,e^2=fg,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, 4574131);