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) = 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 | |
1294 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)(9,10,14,13)(11,15,12,16)(18,22)(20,27)(23,28)(26,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,45)(43,44)(47,48);(9,11,14,12)(10,15,13,16)(18,22)(19,25)(24,29)(26,31)(34,38)(39,45)(40,43)(46,47);(9,12,14,11)(10,15,13,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34)(35,39)(36,43)(37,38)(40,44)(41,47)(42,45)(46,48);(1,3,5,8)(2,6,4,7)(9,13,14,10)(11,15,12,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,35,37,42)(34,39,38,45)(36,41,44,48)(40,46,43,47);(1,4,5,2)(3,7,8,6)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,36)(34,40)(35,41)(37,44)(38,43)(39,46)(42,48)(45,47)Polycyclic presentation:
2^9|a^2=fghi,b^2=fgh,c^2=fgh,d^2=ghi,e^2=fhi,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, 4767955);