Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×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 | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(12,16)(14,15)(18,22)(23,28)(24,29)(30,32)(33,34)(35,36)(37,45)(38,39)(40,43)(41,47)(42,48)(44,46);(3,8)(5,7)(9,10,13,11)(12,14,16,15)(19,25)(23,28)(26,31)(30,32)(34,39)(36,40)(41,45)(44,48);(1,3)(2,5)(4,8)(6,7)(9,11,13,10)(12,15,16,14)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35)(34,40)(36,39)(37,46)(38,43)(41,44)(42,47)(45,48);(1,3)(2,5)(4,8)(6,7)(9,12,13,16)(10,14,11,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36,38,40)(34,35,39,43)(37,44,47,48)(41,42,45,46);(1,2)(3,5)(4,6)(7,8)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,47)(34,41,39,45)(35,42,43,46)(36,44,40,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=fgi,d^2=fhi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4637032);