Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1660 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)(5,6,10,8)(7,12,9,11)(13,14,17,18)(15,21,22,19)(16,20,24,25)(23,27,28,26)(29,30,34,32)(31,35,37,38)(33,40,41,36)(39,44,43,42);(5,6,10,8)(7,11,9,12)(14,18)(15,22)(16,24)(26,27)(31,37)(35,38)(39,43)(42,44);(5,7,10,9)(6,11,8,12)(13,14)(15,19)(16,25)(17,18)(20,24)(21,22)(23,27)(26,28)(29,31,34,37)(30,35,32,38)(33,39,41,43)(36,42,40,44);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,15,17,22)(14,19,18,21)(16,23,24,28)(20,26,25,27)(29,32,34,30)(31,38,37,35)(33,40,41,36)(39,44,43,42);(5,9,10,7)(6,11,8,12)(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28)(29,33,34,41)(30,36,32,40)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=gh,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5135030);