Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(17,18,21,24)(19,25,26,22)(20,28,29,23)(27,30,32,31)(33,34,37,38)(35,41,43,39)(36,44,45,40)(42,47,46,48);(1,2,5,4)(3,6,8,7)(10,13)(14,16)(19,26)(22,25)(27,32)(30,31)(34,38)(36,45)(39,41)(42,46);(9,10,12,13)(11,14,15,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,35)(34,39)(36,46)(37,43)(38,41)(40,48)(42,45)(44,47);(1,3,5,8)(2,6,4,7)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,31)(29,32)(33,36,37,45)(34,40,38,44)(35,42,43,46)(39,47,41,48);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,45)(46,48)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=h,d^2=fgi,e^2=fhi,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, 1719222);