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 | |
1654 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)(13,14)(15,21)(16,18)(17,24)(19,23)(20,26)(22,27)(25,28)(29,30,35,39)(31,36,32,42)(33,34,38,37)(40,44,43,41);(6,9)(10,12)(14,16)(19,21)(20,24)(27,28)(29,31,35,32)(30,36,39,42)(33,43,38,40)(34,41,37,44);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,21)(17,25)(18,23)(20,28)(22,26)(24,27)(29,32,35,31)(30,36,39,42)(33,43,38,40)(34,44,37,41);(1,2,3,4)(13,16,18,14)(15,21,23,19)(17,24,26,20)(22,28,25,27)(29,33,35,38)(30,37,39,34)(31,40,32,43)(36,41,42,44);(5,7,8,11)(6,10,9,12)(13,17)(14,20)(15,22)(16,24)(18,26)(19,27)(21,28)(23,25)(29,34,35,37)(30,38,39,33)(31,41,32,44)(36,40,42,43)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fg,d^2=i,e^2=fgh,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, 4576531);