Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,7)(8,10,12,11)(13,14,17,18)(15,21,23,19)(16,24,25,20)(22,27,26,28)(29,30,33,36)(31,37,38,34)(32,40,41,35)(39,42,44,43);(5,7,9,6)(8,10,12,11)(14,18)(16,25)(19,21)(22,26)(31,38)(32,41)(34,37)(35,40);(5,8,9,12)(6,10,7,11)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,27)(26,28)(29,31)(30,34)(32,39)(33,38)(35,42)(36,37)(40,43)(41,44);(1,2)(3,4)(13,15,17,23)(14,19,18,21)(16,26,25,22)(20,28,24,27)(29,30)(31,37)(32,35)(33,36)(34,38)(39,43)(40,41)(42,44);(13,16,17,25)(14,20,18,24)(15,22,23,26)(19,27,21,28)(29,32)(30,35)(31,39)(33,41)(34,42)(36,40)(37,43)(38,44)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4044305);