Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,7)(8,11,12,10)(14,18)(16,23)(19,24)(22,27)(29,30)(31,34)(32,35)(33,38)(36,41)(37,39)(40,43)(42,44);(2,4)(5,6,9,7)(8,10,12,11)(14,18)(15,21)(20,25)(22,27)(30,31)(32,39)(33,41)(42,43);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,34,30)(32,35,39,37)(33,38,41,36)(40,42,44,43);(5,8)(6,10)(7,11)(9,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32,34,39)(30,35,31,37)(33,40,41,44)(36,42,38,43);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1712924);