Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1758 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)(5,6,9,7)(8,11,12,10)(13,14)(15,21)(16,24)(17,18)(19,22)(20,25)(23,26)(27,28)(29,30,34,32)(31,37,38,35)(33,40,41,36)(39,43,42,44);(5,6,9,7)(8,11,12,10)(14,18)(15,22)(20,24)(23,28)(30,32)(31,38)(33,41)(43,44);(6,7)(10,11)(13,14)(15,21)(16,24)(17,18)(19,22)(20,25)(23,26)(27,28)(29,31,34,38)(30,35,32,37)(33,42,41,39)(36,44,40,43);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,23)(17,22)(18,21)(20,26)(24,27)(25,28)(29,32,34,30)(31,37,38,35)(33,40,41,36)(39,44,42,43);(5,8)(6,10)(7,11)(9,12)(13,16,17,25)(14,20,18,24)(15,23,22,28)(19,26,21,27)(29,33,34,41)(30,36,32,40)(31,39,38,42)(35,43,37,44)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fgh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 4944395);