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 | |
1596 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)(7,8)(13,14,18,21)(15,22,20,17)(16,24,27,19)(23,26,25,28)(30,34)(31,37)(36,40)(39,43);(5,6)(7,11)(8,12)(9,10)(15,20)(17,22)(23,25)(26,28)(29,30,33,34)(31,38,37,35)(32,40,41,36)(39,42,43,44);(5,7,10,8)(6,11,9,12)(13,15,18,20)(14,17,21,22)(16,25,27,23)(19,28,24,26)(29,30)(31,35)(32,40)(33,34)(36,41)(37,38)(39,44)(42,43);(1,2,3,4)(5,8,10,7)(6,11,9,12)(13,16,18,27)(14,19,21,24)(15,23,20,25)(17,26,22,28)(29,31,33,37)(30,35,34,38)(32,39,41,43)(36,42,40,44);(5,9,10,6)(7,11,8,12)(13,17)(14,20)(15,21)(16,26)(18,22)(19,25)(23,24)(27,28)(29,32,33,41)(30,36,34,40)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=fh,d^2=i,e^2=f,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, 5187538);