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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1939 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)(5,6,8,10)(7,9,11,12)(13,14,17,18)(15,21,22,19)(16,24,25,20)(23,26,28,27)(29,30,33,37)(31,35,39,34)(32,36,41,44)(38,43,40,42);(7,11)(9,12)(14,18)(15,22)(16,25)(26,27)(29,31,33,39)(30,34,37,35)(32,40,41,38)(36,43,44,42);(5,7)(6,9)(8,11)(10,12)(13,15,17,22)(14,19,18,21)(16,23,25,28)(20,26,24,27)(29,31)(30,35)(32,38)(33,39)(34,37)(36,43)(40,41)(42,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,21)(16,20)(17,18)(19,22)(23,27)(24,25)(26,28)(29,30)(31,34)(32,36)(33,37)(35,39)(38,42)(40,43)(41,44);(13,16,17,25)(14,20,18,24)(15,23,22,28)(19,26,21,27)(29,32,33,41)(30,36,37,44)(31,38,39,40)(34,42,35,43)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=g,e^2=f,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, 1545522);